# Full text of "The Elements of geometrie of the most auncient philosopher Evclide of Megara"

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The Dihner Library of the History of Science and Technology SMITHSONIAN INSTITUTION LIBRARIES * — i* •• . ., V <5- .:') -., OU^~<t feVvt, -cfv i-M.J-t> O'-'-'-’ ■ '■■\ i'£.-e- V ‘V- $ Ail- -'CLL^s^ it / /» i ' f \ I '<;■ f ■ .4, 3.-rt nAj^sL/Mj"' -\i -A, Bb Marinas Talomcus J^ms\|, mwassg THE ELEMENTS OF GHOMHTRIH of the moll aunci- enc Philolopher EFCLIDE ofMeaara, Strabo [jtralus Faithfully ( now firsl ) tran* fated into the Englife toung , by H. BillingWiy ,Cuiz.en of London, Whereanto are annexed certaine Scholtes , Annotations, and Indenti¬ ons, of the ieff Mathematici¬ ans, both of time paft , and in this our age , cPoltbt j lljfarcfms R-tisssssns^^ss \ With a very fir uitf nil Preface made by M. I. Dee, ; jpecifying the chief if1! athematicali Scieces, Vehat \ thej are, and vrherunto commodious -.where, alfo, are s difclo fed certaine new Secrets Mathematicall \ and McchanicallswitiU theft our dates, greatly miffed. Mtronomia Geometria 'u\mi Mu sica Anthmctica illL \\yi( H Y.1‘ l'Yfi 'Y , a v Wu nV/JSAVi 00 0 ® isi ywfyf/ Imprinted at London by holm ID aye. 5* The T ranikior to the Reader. Here is (gentle Trader') nothing ( the wordoffod oncly fet apart') which fo much beautifieth and a- dorneth the foule^ and minde^ of ma. as doth the knowledge 0 andfciences: as th ye ofnaturalland moraU'Ttoi- ojophie . The onefetteth before our eyes. the^ creatures offfod. : in which as both in in o^glafe. we beholde the exceding maiefiie andwifedome of fjod. in adorning and beautifying them as we fee : ingeuingyn - to them fuch wonder full and mani folde proprieties . and naturall workjnges. and that fodiuerflyand in fuck yarietie : farther in maintaining and conferuing them continually .whereby to praife and adore him . as by SIPaule we are taugh t * The other tea t- cheth ys rules and ' preceptes of yertue.how yin common life'4-j- mongef men u, we ought to walke yprightly :what dueties per- taine to our felues. whatpertaine to the gouernment or good or** der both of an houfholde. and alfo of a citie or commonwealth. The reading lihpwife ofhifiories.conduceth not a htle.to the ad¬ orning of the foule & minde of man .a fiudie of all men comen¬ ded: by it are feene and fnowen the artes and doinges of infinite wife men gone before ys . In hifiories are contained infinite ex¬ amples ofheroicall yertues to be of ys followed. and horrible ex¬ amples offices to be ofys efchewed . oZAdany other artes alfo there are which beautifie the minde of man: but of all other none do more garnifhe & beautifie it. then thofe artes which are cal¬ led Ala t hematic a ll . Unto the knowledge of which no man can attaine .without the perfeBe knowledge md infiruBion of the principles .groundes .and Elementes cf Geometric , Hut per- y 83s* H* feBly $& TheTranflator to the Reader, feBly to be injlruBed in them, requireth diligent fludie and rest * dingof olde auncient authors . fAmongefl which, none for a be¬ ginner is to be preferred before the mofl auncient Fhilofopher Euclide o/TVlegara . For of all others he hath in a truc^ me¬ thods and infle order, gathered together whatfoeuer any before himhad ofthefe Element es written: inuenting alfo and adding mar\y t hinges of his owne : wherby he hath in due forme accom¬ pli fhed the artefrfgeuing definitions principles, & ground es , wherofhe deduceth his F r op o fit ions or conclufions fnfuch won - derfullwife, that that which goeth before , isofnecefitie requi¬ red to theprouf-j t f that which follow eth . So thatwithout the diligent fludie of Euclides Element es, it is impofiible to attaine •vnto the perfette knowledge of Cfcometrie, and confequently of any of the other Math ematic all jciences . Wherefore confide - ring the want & lacfe offuchgood authors hitherto in our Eng- hfhe lounge, lamenting alfo the negligence, and lachp of ge alts to their countrey inthofe of our nation , to whom Cfod hath geuen loth knowledge, & alfo abilitie to tranflate into our tounge,and to pub lifhe abroad fuch good authors, and booses ( the chiefe in- flrumentesofalllearninges') : feing moreouer that many good wittes both of gentlemen and of others of all degrees, much de- firousandjludiousofthefe artes, and feeling for them as much as they cm, faring no games, andyetfrufirate of their intent, by no meancs attaining to that which they feekgs : I haue^> for their fakes, with fome charge & great trauaile, faithfully tran- Jlated into ourtulgare touge,(yfet abroad in Frint , this booke ^Euclide. Whereunto I haue added eafie and plaine decla¬ rations and examples by figures, of the definitions . In which bookp alfoye fhall in due place finde manifolde additions, Scho- lies. Annotations, and Inuentions : which I haue gathered out of many of the mofl famous & chiefe Mathematics s , both of old time, and in our age: as by diligent reading it in courfe,ye fhall well m S-i/The Tranflater to the Reader, Well perceaue . Fhe fruite and game which I require for theft my paines and trauailefeall be nothing els, but onely that thou gentle reader , will gratefully accept the fame^ : and that thou may eft thereby receaue jome profite:and moreouer to excite and fir re vp others learned \ to do the like > to take paines in that behalfe . 'By meanes wherofour Bnghfhe tounge fall no lejfe be enriched with good Authors > then are other jlraunge tounge s: as the Butch , French , ftalian , and Spanifhe : in which are red all good authors in a maner, found amongef the (ftrekes or Latines . Which is the chief ef caufe , that amongef the do flo-* rijhe fo many cunning and fkilfull men , in the inuentions of fraunge and wonderfull t hinges, as in thefe ourdaies we fee there do . Which fruite andgaine if I attaine Dntoy itjhall encourage me hereafter, in fuch like fort to tranfate , and fet abroad feme other good authors, both pertaining to religion ( as partly I haue already done ) and alfo pertaining to the <£\d'athe~ maticall Artes . Thus gentle reader farewell* 00 a*4 ■ »t - ' IT? - ■ s- ;,"V- - , w- . ■;■■■■ V. •: f - ' J - . \ . V . . - . V ■ 'ijjs ■ -- •> ■ • -.V * o i .* v*. > • * ' . ‘ \ - . ; " : \ ; • .. ' :: ; V T.'.. : i ) • , . . \ • '■ '■ ' . , . . ' - . i . . \ . T. .>r.\ ^ii-y,v3 ■ ■ TMiOVvow; s -\\\ovr^ . \ s ..y ssk , _ . \\ y\; ■ r . / . . i vr, 7;" . ,-V-: % * i . . 2 & - ti \ . i y '• -E i \ ■ ~4 \ . . - . 5 ■ . - s ■ jg q &i : ' - ■ ■ ■■■ '■ i . . ■ - ' . - , ' v ... .■ ' V . \ . . 'so mow./,1 -as}. ' • o, •. '*Avv\\Va\ \mu>vovi!fv.* o\ ,1\V \ * ' ■ ' i o \ . ■ " . 0; \ . \ n c \w\\ " ' % /. / i V- ■ . 1 './V. • \\ . ^vVvi:l‘ 4; 0 , o'Ui, ' , . ? S'.'V . *r*\ .f\ ' . . : ■* ■■■•- * . w . . . - ■ . • ■ ’!■ ■ ■ '• 5**>TO the vnfained lovers of truthe , and conftant Studentes of Noble Sciences O H N DEE of London Jhartilj wifheth grace from heauen, and moft profpe- rous JucceJJe in all their honejl attempt es emd exercifes* luine TlatOy the great M after of many worthy Philofophers* and the conftant auoucher, and pithy perfwader of Vnurn * Bo- nttm , and Ens : in his Schole and Academic, fundry times (befides his ordinary Scholers) was vifited of a certaine kinde of men, allured by the noble fame of Plato, and the great commendation of hys profound and profitable doctrine* But when fuch Hearers,after long harkening to him, perceaued, that the drift of his difcourfes ifftied out, to conclude , this Vnttm , Bo* num ,and Ens-, to be Spirituall,Infi- nite, iEternall , Omnipotent , &c. Nothyngbeyngalledged or exprefled,How, worldly goods: how, worldly digni- tie:how,health,Stregth or luftines of bodyrnor yet the meanes,how a merueilous fenfibleand bodyly blyfte and felicitie hereafter, might beatteyned: Straightway* the fantafies of thofe hearers,were dampt: their opinion ofjP/^t^was dene chaun- ged:yea his dodrine was by them defpifed:and his fchole , no more of them vifi- ted.Which thing,his S choler, Arijlotle, narrowly cofictering/ounde the caufe ther- of,to be, For that they had no forwarnyng and information,in general! , whereto >t> his dodrine tended.For,fo,might they hauc had occafion, either to haue forborne his fchole hauntyng : (if they, then, had mifliked his Scope and purpofe ) or con- ftandy to haue continued therinrto their full fatiftadion : if fuch his finallfcope& intent , had ben to their defire . Wherfore* Arijlotle, after that,vfed in brief,to forewarne his owne Scholers and hearers , both of what matter > and alfo to what ende,he tooke in hand to fpeake, or teach . While I confider the diuerfe trades of ,j* thefe two excellent Philofbphers ( and am moft fure,both, that Plato right well, o- therwife could teach : and that (^Arijlotle mought boldely , with his hearers , haue dealt in like forte as Plato did)I am in no little pang of perplexitie : Bycaufe , that, which I miflike,is moft eafy forme to performe (and to haue Plato for my exaple.) And that, which I know to be moft commendable: and (in this firft bringyng,into common handling, the <_ Antes CEtathematicall) to be moft neceftary : is full of great difficultie and fundry daungers.Y et, neither do I think it mete, for fo ftraunge mat- ter(as now is ment to be publifhed)and to fo ftraunge an audience , to be bluntly, at firft, put forth, without a peculiar Preface : Nor (Imitatyng Arijlotle) well cart I hope , that accordyng to the amplenes and dignitie of the State CMathematicall , I am able, either playnly to prefcribe the materiall boundes : or precifely to expreftc the chief purpofes, and moft wonderfull applications therof. And though lam fure , that fuch as didfhrinke from Plato his fchole , after they had perceiued his ft- lolin D^e his Mathematical! Preface. ilall conclufion, would in thefe thinges haue ben his moft diligent hearers ) fo infi¬ nitely mought their defires, in fine and at length , by our Artes Mathematical! be fa- tifHed)yet,by this my Preface & fbrewarnyng , Afwell all fuchgiiay (to their great behofe)the loner, hither be alluredras alfo the Pphagoricalf and Platomcall perfect fcholer,and the conftant profound Philofopher, with more eafeand fpede,may (like the Bee,) gather, hereby,both wax and hony. Wherfore, fey ng I finde great occafion(for the caufes alleged, and fafder, in re- 5? fped of my Art CMathcmatike general! ) to vfe a certaine forewarnyng and Preface, » whofe content flialbe,that mighty, moft plefaunt,and frutefull Mathematical!, Tree , The intent of with his chief armes and fecond(grifted)braunches: Both, what euery one is , and this Preface . alfo, what commodity, in general!, is to be looked for,afwelI of griff as ftockeAnd „ forafmuch as this enterprife is fo great, that, to this our tyme , it neuer was (to my ,, knowledge) by any achieued : And alfo it is moft hard , in thefe our drefy dayes, to fuch rare and ftraunge Artes, to wyn due and common credit : N euertheles , if, for my fincere endeuour to fatiffie your honeft expectation , you will but lend me your thakefull mynde a while:and,to fuch matter as,for this time, my penne (with fpede)is hable to deliuer, apply your eye or eare attentifely : perchaunce , at once, and for the firft falutyng,this Preface you will finde a leffon long enough. And ei¬ ther you will, for a fecond ( by this ) be made much the apter: or fhortly become, well hable your feiucs, of the lyons claw , to coniedtire his royall fymmetrie , and farderpropertie . Now then, gentle, my frendes, and coufitrey men,Turne your eyes, and bend your myndes to that dbdrine , which for our prefent purpofe , my fimple talent is hable to veld you. ^11 thinges which are,& haue beyng, are found vnder a triple diuerfi tie generall. For, either, they are demed Supematurall,Naturall,or,of a third being.Thinges Supematurall, are immatcriall, fimple, indiuifible, incorruptible, & vn changeable. Things Naturall, are materiah,compounded,diuifiblc, corruptible, and chaungea- ble.Thinges Supernatural], are, of the minde onely,cOmprehended:Things Natu¬ rall, of the fenfe exterior, ar hable to be perceiued.In thinges Naturall, probability and coniedure hath place: But in things Supematurall, chief demoftration,& mof| fure Science is to be had. By which properties & comparafons of thefe two, more eafilv may be defcribed,the ftate, condition, nature and property of thofe thinges, which, we before termed of a third being: which, by a peculier name alfo,are called Thjnges at hematic ail. F or,th efc,bey ng (in a maner)middle, betwene thinges fu- pernaturall and naturalharc not fo abfolute and excellent, as thinges fupernaturah Nor yet fo bafe and groffe,as things naturall: But are thinges immateriall : and ne- uertheleffc,by materiall things hable fomewhat to be fignified . And though their particular Images , by Art,areaggt egableand diuilible : yet the generall Formes^ nouvithftandyng,areconftant,vnchaungeable,vntrafformable,andincorruptible. Neither of the fenfe, can they, at any tyme, be perceiued or iudged.Nor yet, for all that, in the royall mynde of man, firft conceiued.But,furmountyng the imperfedio o f c on i edur e, weeny n g and opinion:and commyng fhort ofhigh teteUcdaall c5- cepti6,are the Mercurial fruite of Dianceticall difcourfe,in perfect imagination fub- fiftyng. Ameruaylousnewtralitiehaue thefe thinges ^Mathematical! . and alfo a ftraunge participate betwene thinges fupematurall,immortall,intelle(5lual, fimple and indiuifible:and thynges naturall, mortall/enfible, compounded and diuifible. Probabilitie and fenfible profe,may well feme in thinges naturall: and is commen¬ dable: In Mathematical! reafoninges,a probable Argument, is nothyng regarded: nor yet the teftimony of fenfe, any whit credited : But onely a perfed demonftra- tion, oftruthes cemine,neceftary,and inuindbkoWniuerfally and neceffaryly con¬ cluded 2 i . ’ • , . •» ' 031 John Dee his MatKematicall Prasfkce* eluded: is allowed as fufficientfor an Argumentexa&Iy and purely Mathematical. Of Mathematical! thingesgre two prin cipall kindcs : namely, Numbagmd Mag- nitnde.Tiumberfve define, to be, a eertayne Mathematical! Sumc}of Fnits. And, an Jjfriity is that thing Mathematical!, Indiuifible , by. participation of fome likenes of whofe property , any thing,which is in deede,oris counted O ne^may refonabiy be called One . We account an Fnit^z. thing Mathematicall , though it be no Number, and alfo indiuifible.'bccaufe^fi^materially^Number doth confifi V which 9 princi¬ pally , is a thing Mathematical l. Magnitude is a thing Mathematicall y by participation of fome likenes of whofe nature , any thing is iudged long , broade, or thicke . A thicke Magnitude we call a Bolide. , m a Body . WhazMagmtudefo euer,i$ Sblide or. Thicke,is alfo broade,& long. A broade magnitude, we call a Superficies or a Plaine, Huery playne magnitudc,hath alfo length. A long magnitude, wetermea Line. A Line is neither thicke nor broade, butonely long : Euery ceitayne Line, hath two endes;The endes of a line, are Pointes called. A Point f s a thing Mathematicall , indb uilible, which may haue a certavne determined fituation . If a Povnt mouefrom a determined fituation , the way wherein it moued, is alfo a Line : mathematically produced, whereupon, of th e auncicn t Mathematicicn s,a Line is called the race or courfe of a Point . A Poynt we define , by the name of a thing Mathematical!} though it be no Magnitude, and indiuifible: becaufeitis thepropre ende, and bound of a Line : which is a true Magnitude . And Magnitude we may define to be that thing Mathematical f which is diuifible for eucr,in partes diuifible, long, broade or thicke . Therefore though a Poynt be no ^Magnitude, yetT erminatiuely weree- ken it a thing Mathematicall (as I layd)by reafon it is properly the end </ and bound, of a line. . *> Number. Note the rt)oricf Unit, to ext refit the Grel>e Mo-~ not Vni- tie : at tee kau* d/l, commonly, till nowjvfed. Magnitude. »> A point, oi Line . N either ‘Number, not OMagnUtudeJiam any Materialitie. Firft,we will confidef of Number,and of the Science Mathematicall , to it appropriate, called Arithmetifoi and afterward of Magnitude fmdhis Science, cdkdGeometrie. But that name con* tenteth me not: whereofa wordor two hereaftcr fhall be fayd . How Immaterial! and free from all matter , Number is y who doth hot perceaueW yea, who dotbndl wonderfully woder at ME or, neither pure Element, nor Ariftoteles, Quinta Ejfcntia, jshable to feme for Number, ashis propre matter. Nor yet the puritie and fimple- nes of Subfiance Spiritual! or Angelicall , will be found propre enough’ thereto. And therefore the great & godly FhilpCofk&i Anitms Boetws, fayd : Omnia qu&cimfp apriffidua rerum natutd cmfiru£htfmty Numemrumvidentur ration? format a. Hoc enim ftpit principal? iri amnio € on dh oris Exemplar . THatfS ^ thinges ( 1 t>hich front the Titery firft qrtgtnall being of ihiiigef0 -Mite bene framed and made) do appear? to be Formed by the reajon of jSlumbers V For this 'toas the princi pall example or patterne in the mind? of the Creator . O comfor¬ table allurcmcnr, O rauilhing perlwafion, to deale with a Science, whofe Subie^ is fo Auncientdo pure,fo exyelienWfc lurmouii ting all creature5,fb yfed of the Al¬ mighty and incomprehenfiblc wifdomc of the Creator5 in the diftimfi creation of allcreatures: in all their diftinbt partes, propertiesAnatures , and vertues, by order, and moft abf0lute number,brought,from Npthingyto the Formalities their being andfiate.By Numbos propertic therefore, ofvs,by all pofiible meanes,(to the per- fe#ion of the Science ) learned, we may both winde and draw our felues into th^ inward and deepe fearch and ve>v0of all creatures diftimfi vertues, natures, proper- tie^, and Formes: And alfojfarderjarifejcfimejalcend^nd mount vp ( with Specula. tiue winges } in fpirit, to behpld in the Gks of Creation, the Forme of femes y the Exemplar Number ofall thing!esiiV^^4^/bpdl vifible and inuifible .• mortall and , 7 - - *♦].. immortal! lohnD ee his Matliematicall Preface . > immortall,C orporall and SpirituallJPart ofthis profound and diuine Science, had Joachim the Rrophefier attey ned vnto : by ‘Numbers Formal!., Natural!,, and Rational!, forfeyng,concludyng,and forfheWyng great particular euents , long before their comming.His bookes yet remainyng, hereof, are good profe: And the noble Earle of Miranduk,(\>&d.zs thar,)a fufficient witnefle:that loachim,in his prophef.es, proce* trim* 1488* 4ed by no other my, then by Numbers Formal!. And this Earle hym felfedn Rome,*fet vp -poo. Conduftons,in all kinde of Sciences, openly to be difputed of :and among the reft, in his Conclufions CMathematicall, (in the eleuenth Conclufton ) hath in Latin,this Englifh fenten cc. By Numbers, a my is had , to thcfearchyng out, dud vnder - jlandyng of euery thyng , hable to he knowen . For the 'verifying of which Conclufon , I pro - mife toaurfvere to the/ 4. glujtflions^mder written, by the way of Numbers .Which Co- clufions,I omit here to rehearle: afwell auoidyng fuperfluousprolixitie:as , by- caufe Joannes Ficus, workes , are commonly had. Bht,in any cafe, I would wifh that thofe Conclufions were red diligently , and perceiued of fuch,as are earn eft Ob- feruers and Confiderers of the conftant law of nubers: which is planted in thyngs Naturall andSupernaturalljandis preferibed to all Creatures, inuiolably to be kept*For,fo,beftdes many other thinges , in thofe Conclufions to be ma'rked,it would apeare,how. fincerely,& withinmy boundes,I difclofe the wonderfull my- fteries,by numbers,to be atteynedvnto. ■ Of my former wordes,eafyit is to be gathered,that Number hath a treble ftatei One, in the Creatorum other in euery Creature(in refped of his complete conftb tution;)aud the third,in Spiritualland Angelicall Myndes,and in the Soule of ma. In the firft and third ^ant,Numher > is termed N timber Numbryng. But in all C rea- tures,otherwife,7V//w^,is termed NuberNumbred. And in our Soule, N uberbea- teth fuch a fwaye^and hath fucit anaftmitie therwith .■ that fome of the old Philofo - pbers t&npftt,Makr$otde,to brgaNumbefmouyng kfitfe: And in dede,in vs, though it beu Yery Accident: yet fuch aii^ Accident it is;thait before all Creatures it had per-; £b<S? beyng, in thc.C vcatorfcmpkdma&y.NumberNumbryng therfore,is the difere- tion difccrning,and. diftincHng ofthinges* But in God the Creator , This dilcre- tion,in thebeginnyng,produ€ed\orderly anddiftimftly allthinges. Eor his Num- foy^,then,was his Creatyngbf all.thinges. Aral his QontmnaWi Numbryng , of all thinges,is the Gonfemation of them in being: And^where and when he will lacke an :Fnit: there and thaivthat particular thyng ^ftialbejDi/c^fez/ddefe I ftay.But our Seumllyng^difti,n'#yng^nd^»i^r)fj^,createth'ii6thyngrbutofMultttudexom fidered,raakedi Gertaine and diftinCtdetermination , And albeit thefe thynges be Whty and .truces. ofgreat , ,yer (.b^tbe infinite gso^esoftheAl- mighty Zemarie,) Artmqall Methods and eafyw^yes are made , by which the ze- Philofopher jiifay wyn nere this fliuerifti iddltMs Moiintay rie of Cohtempk- tionrand momthen^Rnitempfatibh.Andallbkhotigh Number, bed. thyng To I in- iV» T i 1 1 f & A frvv-l 4 1 1* t rnf'. Ka r rl IT v tT Vi m 1 1 A ft KF lower, to thynges fenftbly pefeeiuedras of a mbmentiinye found e iterated: then td the lcaft thynges that may bc'fCen,niimerable:Arid at length, (moff groffely,) to a multitude of any corporall thynges feeh,or felt: and fb,of thefe grofte and fenftble thynges, we are trayhed to leame a ce'rtaine Image or likenes of numbers : and to vfe Arte in them to our pleafufe arid proffit.So grofte is our conuerfation, and dull iSourapprehenfion: while mortal! Senfe, in vs, ruleth the common wealth of our litle world.Hereby we fay, Three Lybns,afe three: or a F ernarie . Three Egles^are three, or a F ernarie • Which* T ernaries , are eche,the Fnion, knot, and Zniformitie,o? three diferete and diftimft Vnit-i* That^ is, we may ineche T ernarie , thrife , feuerafty pointe,and fhew a pan, One, One, and One. Where, in Numbryng, we fay One, two. Three. John Dee his Mathematical! Preface. Three . But how farre, thefe vifible Ones , do differre from our Indiuifible Vhits (in pure Arithmetike, principally confidered)no man is ignorant . Yet from thefe groife and materiall thynges,may we be led vp ward, by degrees, fofinformyng our rude Imagination,toward the coceiuyng of Numbers/dbfoluttly ( :N ot fuppoling, nor admixtyng any thyng created,Corporall or Spiritual!, to fupport,conteyne,or reprefent thole Numbers imagined : ) that at length, we may be hable , to finde the number of our ownc name , glorioully exemplified and regiftred in the booke of the Trinitk moft blelfed and asternall. But farder vnderftand,that vulgar PraCtifers,haue N umbers otherwife, in fun- dry Confiderations:and extend their name farder, then to N umbers , whole leaft partis an Vnit .F or the common Logili,R cckenmaftcr, or Arithmeticien, in hys v- fing of N umbers : of an V nit,imagineth lelfe partes : an dcalleth them Fractions. As of an Vnit , he maketh an halfe,and thus noteth it, A., and fo of other, (infinitely di¬ uerfe) partes of an Vnit. Yea and farder, hath, Fractions of Fractions. &c . And,foraf much, as, Addition > Subfir action , Multiplication , Dim [ion and Extraction of Rotes, are the chief, and fufficient partes of Ar Ahmet ike .* which is , the Science that demon fir a* teth the properties, of Numbers, and all ope ratios , in numbers to be performed /How often, therfore, thefe fiue fundry fortes of Operations, do , for the moll part, of their cxc- „ Note, cution,dilfcrre from the fiue operations oflike generall property and namefin our ,, Whole numbers praCtilable,So often , (for a more difiinCt doctrine ) we, vulgarly „ accountand name it, an other kynde of Anthmetike .And by this realon.-the Com fideration,dodtrine,and working, in whole numbers onely: where, ofan Vnit, is no Idle part to be allowed: is named (as it were)an Arithmetike by it lelfe . And fo of the Arithmetike of Fractions. In lyke forte, the neceflary,wonderfull and Secret doc¬ trine of Proportion , and proportionalytie hath purchafed vnto it lelfe a peculier 2, maner of handlyng and workyng:and fo may feme an other forme of Arithmetike. Moreouer,the Astronomers for fpede and more commodious calculation,haue de- 2 . uifed a peculier maner of orderyng n fibers, about theyr circular motions, by Sexa- 3 genes, and Sexagefmes.By Signes, Degrees and Minutes &c . which commonly is called the Arithmetike of Agronomical or Phificall Fr actions. That, haue I briefly no¬ tedly the name of Arithmetike Circular. Bycaufe it is alfo vfed in circles,nc t Afiro- wmicall. c. Pradtile hath led Numbers farder , and hath framed them, to take vpon A. . them , the fhew of Magnitudes propertie: Which is Incommenfurabilitie and Irratio - nalitie. (For in pure Arithmetike, cm Vnit, is the common Meafurc of all N umbers.) A nd,here,N fibers are become, as Lynes,Playnes and Solides: fome tymes Ratio- nail, fome tymes Irrationall. And haue propre and peculier characters, (as c£. and fo of other. Which is to lignifie Rote Square , Rote Cubik:and fo forth: )& propre and peculier falhions in the fiue principall partes: Wherfore the pradifer,e!temeth this,a diuerfe Arithmetike from the other . PraCtife bryngeth in,here,diuerfe com- poundyngofNumbers: asfome tyme, two, three, foure(or more) Radicall nubers5 aiuerlly knit, by lignes, ofMore & Lelfe:as thus 12 + v'ct i5.0r thus ig -b v/cc'i^—bb‘2. &c. And fome tyme with whole numbers, orfraCtions of whole Number,amog them : as 20 33-V^ xo. — -j-CcCp. And fo, infinitely , may hap the varietie. After this : Both the one and4 the other hath fractions incident:andfo is this Arithmetike greately enlarged, by diuerfe ex- hibityng and vfe of Compofitions and.mixtyriges . Confidcr how, I (beyng deli- rous to deliuer the ftudentfrom error and Cauillation)do giue to this PraCtife, the name of the Anthmetike of Radicall numbers: Not, of Irrationall or Surd Numbers; which otherwhile, are Rationall : though they haue the Signeofa Rote before b *.ij* them., lohn Dee his Mathematical! Preface. them, which, Arithmetike of whole Numbers moft vfuall , would lay they had no fuch Roote: and fo account them Surd Numbers: which, generally (poke, is vntrue; as Euclides tenth booke may teach you. Therfore to call them , generally , Radical! Numbers, (by reafon of the ligne / .prefixed,) is a fure way : and a diffident generall diftinftion from ail other ordryng and vfing of N umbers : And yet ( befide all this)Confider : the infinite delire of knowledge , and incredible power of mans Search and Capacitye: how, they, ioyntly haue waded farder ( by mixtyng offpe- culation and praftife)and haue found out , and atteyned to the very chief perfec¬ tion (almoft) of Numbers Pradicali vfe. Which thing,is well to be perceiued in that great Arithmetical! Arte of Aquation : commonly called the Rule of Cojf. or ^Alge¬ bra. The Latines termed it,Regulam Ret & Cenfus , that is , the rftile of the thyng and his lvalue* With an apt name : comprehendy ng the firft and laft pointes of the worke . And the vulgar names , both in Italian , Frenche and Spanilh,depend(in namyng it,) vpon the lignification of the Latin word,i?«: A f/;/wg:vnleaft they vfe the name of ^Algebra. And therm (commonly)is a dubble crror.The one,of them, which thinke it to be of Geber his inuentyng: the other of fuchascallit Algebra. For, firft, though Geber for his great fkill in N umbers, Geometry, Aftronomy , and other maruailous Artes,moughthauefemed hable to haue firft deuifedthefayd Rule: and alfo the name carryeth with it a very nere likenes of Geber his name : yet true it is, that a Greke Philofopher and Mathematicien,named Diophantus, before Geber his tyme,wrote i3.bookes therof ( of which , fix arc yet extant : and I had *Anno,is^t3, them to * vfe,of the famous Mathematicien,and my great frende, Petrus <j\€ont. au¬ reus : ) And fecondly,the very name, is Algiebar, and not Algebra : as by the Arabien Autcen, may be proued: who hath thele precife wordes in Latine,by Andreas Alpa~ ^tffmoft perfeft in the Arabik tung ) fo tranflated . Scientia faciendi Algiebar & Almacbabel. i. Scientia inueniendi numerum ignotum,ptr additionem Numeri , & diuifeo . nem & aquationem. Which is to fay:T he Science of^oorhyng Algiebar and yfl* machabelfhdzisjhc Science offindyng an Imknowen number , by Addyng of a Number, er Diuijton aquationMcrc haue you the name : and alfo the prin- cipall partes of th e R u!e,touched.T o name it0The rule, or Art of Aquation, doth fig- nifie the middle part and the State of the Rule . This Rule, hath his peculier Cha- rafters: and the principal partes of Arithmetike, to it appertayning,do differre from the other Arithmetical! operations. This Arithmetike, hath Nubers Simple, Copound, Mixtrand Fraftions, accordingly. This Rule, and Arithmetike of Algiebar, is fo pro¬ found, fo generall and fo (in maner) conteynedi thewhole power of Numbers Application prafti call: that mans witt, can deale with nothyng,more proffitable a- bout numbers : nor match , with a thyng , more mete for the diuine force of the Soule, (in humane Studies, affaires, or exercifesjto be tryedin. Perchaunceyou looked for, (long ere now,) to haue had fome particular profe, or euident teftimo- ny of the vfe, profit t and C ommodity of Arithmetike vulgar, in the C oiftmon lyfe and trade of men.Therto,then,I will now frame my felfe : But herein great care I haue, leaft length of fundryprofes, might make you deme, that eitherl did mi£ doute your zelous mynde to vertues fcliole : or els miftruftyour hable witts , by fome,to geffe much more. A profe then,foure,fiue,or fix, fuch , will I bryng , as any reafonable man,thenvith may be perfuaded,to loue & honor , yea karne and exercife the excellent Science of Arithmetike. And firft: who, nerer at hand, can be a better witnefte of the frute receiued by Arithmetike, Aon all kynde ofMarchants i Though not all, alike, either nede it,or vfe it.How could they forbeare the vfe and helpe of the Rule , called the Golden Rule? Iohn Dee his Mathematical! Preface. Rulec’Simple and Compounde:both forward and backward < How might they miffe Arithmetkall helpe in the Rules ofFelowfhyp: either without tyme, or with tyme'and betwene the Marchant& his Fa&or i The Rules ofBartering in wares onely: or part in wares, and part in money, would they gladly want i Our Mar- chant venturers, and Trauaylersouer Sea , how could they order their doynges iuftly and without Ioffe , vnlcaft ccrtainc and generall Rules for Exchauge of mo¬ ney 5and Rechaunge, Were, for their vfe,deuifed f TheRuleofAlligation,in how fundry cafes,doth it conclude for them,fuch precife verities, as neither by natural! witt , nor other experience, they, were hable, els, to know < And ( with the Mar- chant then to make an end ) how ample & wonderfull is the R ule of Falfe pofiti- ons < efpecially as it is now, by two excellent Mathematiciens ( of my familier ac- quayntance in their life time ) enlarged < I meane Gemma Frifius, and Simon Jacob . Who can either in brief conclude , the generall and Capitall R ulest or who can I- magine the Mvriades of fundry Cafes,and particular examples,in Ad and earn eft, continually wrought, tried and concluded by the forenamed Rules, onely < How fundry other Arithmeticall praclifes , are commonly in Marchantes handes,and knowledge: They them felues,can,at large, teftifie. The Mintmafter,and Goldfinith,in their Mixture of Metals , either of diuerfe kindes,qr diuerfe values.-how are they, or may they,exadly be directed , and mer- uailoufly pleafured,if Arithmetike be their guided And the honorable Phificias, will gladly confeffe them felues much beholding to the Science of Arithmetike^ and that fundry way es : But chiefly in their Art of Graduation , and compounde Medicines, And though Galenas, Auerrou,Armldiis , Lidias , and other haue pu¬ blished their pofitions , afwell in the quantities of the Degrees aboue Tempera* ment , as in the Rules , concluding the new Forme refulting : yet a more precife, commodious, and eafy (JMethodf* extantrby a Countreyman of ours ('aboue 200. yeares ago)inuented. And forafmuch as I am vncertaine , who hath the fame: or when that litle Latin treatife, (as the Author writ it, ) fhall come to be Printed: (Both to declare the defire I haue to pleafure my Countrey,wherin I may : and al- fo,for very good profe of Numbers vfe,in this moftfubtile and frutefull , Philofo- phicall Conclufion, ) I entend in the meane while , moft briefly,and with my far- der helpe, to communicate the pith therof vnto you. Firft defcribe a circle : whole diameter let be an inch . Diuide the Circumfe¬ rence into foure equall partes. Fro the Center, by thofe 4.fedions,extend fright lines : eche of ^.inches and a halfe long : or of as many as you Iifte,aboue 4. with¬ out the circumference of the circle : S o that they fhall be of 4-inches long ( at the leaft) without the Circle . Make good euident markes,at euery inches end. If yott lift, you may fubdiuide the inches againe into 10. or i2.fmallerpartes,equall. At the endesofthe lines, write the names of the 4. principall elementall Qualities. Hote and Colde , one againft the other . And likewife CMoyst and Dry, one againft the other. And in the Circle write Temperate. Which T emperature hath a good La¬ titude : as appeareth by the Complexion of man . And therefore we haue allow¬ ed vnto it, the forefay d Circle : and not a point Mathematicali or Phyficalb Now, when you haue two thinges Mifcible , whole degrees are * truely knowen : Ofneceflitie, either they are of one Quantitie and waight, or of diuerfe. If they be of one Quantitie and waight: whether their formes,be Contrary Qua- lities, or of one kinde (but of diuerfe intentions and degrees) or a T emperate, and a Contrary , T be forme resulting of their Mixture, is in the Middle hetwene the degrees of the R» *Takefom part of LuUm counptylein his booke de QJEjJentk. lohn Dec his Mathematical! Preface. the formes mixt . As for example, let o*f,be Moitt in the firft degree : and B , Dry in the third degree . Adde i. and 3. that maketh 4 : the halfe or middle of 4.1s 2. *No;e, T his 2.1s the middle, equally diftant from A and B ( for the * T emper ament is coun~ ted none . And for it, you mud put a Ciphre, if at any time, it be in mixture). HOTE Counting then from B, 2. degrees , toward you finde it to be Dry In the firft degree : So is the Form refulting of the Mixture of ^,and B , in our example. I will sreue you an other example . Suppofe, you haue two thinges,as C,and D and of C, the Heate to be in the 4.degree : and of D, the Colde, to be remifle,euen vnto the Temperament . N ow,for C,you take 4: and for D,you take a Ciphre h which, added vnto 4, yeldeth one!y4.The middle, or halfe, whereof, is 2. Wherefore the Forme refilling of C , and D, is Hote in the fecond degree: for, 2. degrees,accoun- ted from C3 toward D , ende iufte in the 2. degree of heate . Of the third ma- nei‘,1 will geue alio an example: which let be this : I haue a liquid Medicine whole Nate. Qualitie ofheate is in the 4-degree exalted : as was C, in the example foregoing: and an other liquid Medicine I haue .• whofe Qualitie, is heate, in the firft degree. O f eche o f thefe, I mixt a like quantitie ,* S ubtrad here, the ldfe fro the more .• and the refidue diuide into two equall partes •• whereof, the one part, either added to the idle, or fubtrafled from the higher degree, doth produce the degree of the Forme I qIui Dee his M atliem aticall P rceface. Forme refulting, by this mixture of C,and E . As,iffrom 4. ye abate 1. there rcfteth 3.thehaifeof3-is iff ; Addeto i.thisiQ- ,* youhaue2-L. » Orfubtradfromq. ' this 1 — .* you haue like wife 2-- remayning . Which declarefh 5 the Formereful- ting, to be He ate, in the middle of the third degree. But if the Quantities of two thingesCommixt, be diuerfe, and the Intend- M TheSe- ons ( of their Formes Mifcible ) be in diuerfe degrees , and heigthes. ( Whether C0„J thofe Formes be of one kinde,or of Contrary kindes, or of a Temperate and a jtuic Contrary , What proportion is of the lejfe quant itie to the greater, the fame fall be of the ?> difference jvhich is betwene the degree of the F orme refulting , and the degree of the greater } f quant itie ofthethingmifcible, to the difference, which is betwene the fame degree of the }j Forme refulting^and the degree of the lefe quantitie . As for example . Let two pound of Liquor be geuen, hote in the4.degree:& one pound of Liquor be geued, bote „ in the third degree . I would gladly know the Forme refulting,in the Mixture of }f thefe two Liquors. Set downe your nubers in order , thus. Now by the rule of Algiebar, haue I deuifed averyeafie, briefe, and generall maner of working in this cafe . Let vs firft, fuppofe that Middle Forme refulting , to be iff : as that Rule teacheth . Andbecaufe (by our Rule, here geuen) as the waight of i .is to 2 : So is the difference betwene 4. (the degree of the greater quantitie ) and 12^ • to the difference betwene 12^ and 3.“ (the degree of the thing, in lefle quatitie. And with all, 12^, being alwayes in a cer- taine middell, betwene the two heigthes or degrees) , For the firft difference, I fet 4 — tfdd md for the fecond, I let iff — 3 . And, now agaide, I fay, as i.is to 2X0 is 4— 1%6 to lXC~~3- Wherforc, ofthefe foure proportional! numbers, the firft and the fourth Multiplied, one by the other,do make as ‘much, as the fecond and the third Multiplied the one by the other . Let thefe Multiplications be made accor¬ dingly . And of the firft and the fourth, we haue iff — 3.andofthe fecond &thc third, 8 — 2%^.Whcrfore , our Equation is betwene iff — 3: : and 8 — 22^. Which may be reduced,according to the Arte of Algicbarras, here, adding 3 .to eche part, geueth the iEquation,thus,i2^—i 1—22^. And yet againe,contra<fting, or Redm cing it : Adde to eche part, 22^: Then haue you gff squall to ii : thus reprefen- ted 32^—11. Wherefore, diuiding n.by 3: the Quotient is 3— : theCa/ovofour iff, Cof, or T hing, firft ftippofed. And that is the heigth, or Intention of the Forme rejtdting: which is, Heate, in two thirdes of the fourth degree : And here I fet the Brew of the workc in conclufion, thus The proufe hereof is eafie-by fubtrading 3 .from 3-^-qefleth . r ff .Subtrade the fame heigth of the Forme refulting, (which is 3—) fro 4;: then reijeth W; •You fee, that cfS; double to ff: as 2 . f. is double to 1. & So ihoujd it be by the: rule here geuen . H ote . As yoU ad¬ ded to ech e part of the Equation, 3 : fo if ye firft added to eche part 2%, it would hand, 32^— 3— .8 . And now. adding to eche part 3 .• you haue (as afore) 32^=11. And though I, here,fpeake dnely of two thyngs Mifcible: and moft common- ly,mo then th rec,fourc,fiue or iix,f &c.)are to be Mixed: (and in one Compound *. iiijf. to l 2. Hote. 4. f. I. Hote. John Dee his Mathematical! Prseface. to be reduced.-& the Forme refultyng of the fame, to fettle the turne)yet thefe Ku~ Nvte, les are fufficientrduely repeated and iterated.In procedyng firft, with any two rand then, with the Forme Refulting,and an other ;& fo forth:For, the laft workc, con¬ clude th the Forme refultyng of them all .-I nede nothing to fpeake, of the Mixture (here fuppofed) what it is.Common Philofophie hath defined it , faying, uw/xtio tjl mifcibilium , alter atorum , per minima coniunclorum,Vnio . Euery word in the de-> finition, is of great importance. I nede not alfo fpend any time, to fhew,how,the other manner of diftributing of degrees, doth agree to thefe Rules. N either nede I of the farder vfe belonging to the Croffe of Graduation (before defcribed)in this place declare, vnto fuch as are capable of that,which I haue all ready fay d. N either yet with examples ipecifie the Manifold varieties , by the forefayd two gene- rall Rules,to be ordered. The witty and Studious, here, haue fufficient: And they which are not hable to atteinc to this, without liuely teaching , and more in parti- ' cular: would haue larger difcourfing,then is mete in this place to be dealt withalh And other(perchaunce)with a proude fhuffe will difdaine this litlerand would be vnthankefull for much more . I,therfore conclude : and wifh fuch as haue modeft and earneft Philofophicall mindes,to laude God highly for thisrand to Meruayie, that the profoundeft and fubrileft point, concerning Mixture of Formes and Quali¬ ties Flat urall, is fo Matchtand maryed with the moflfimpIe,eafie,and fbort way of the noble Rule of Algiebar. Who can remaine , therfore vnperfuaded,to louc,a- low,and honor the excellent Science of' Arithmetike * For, here, you may perceiue that the litle finger of Arithmetike, is of more might and contriuing,then ahun- derd thoufand mens wittes,of the middle forte , are hable to perfourme, or truely to conclude, with out helpe thereof. Now will wefarder,by the wife and valiant Capitaine,be certified, what helpe he hath, by the Rules of Arithmetike /in one of the Artes to him appertaining; And Taxi ix?. „ of the Grckes named TcwlixiThat is , the Skill of Ordring Souldiers in Battell ray 5, after the beft maner to all purpofes.This Art fo much dependeth vppon N umbers vfe,and the Mathematicals, that J&lianus ( the beft writer therof, ) in his worke,to the Emperour Hadrianus , by his perfection, in the Mathematicals,(beyng greater, then other before him had,) thinketh hisbooke topafle all other the excellent workes,written of that Art,vnto his dayes.For,ofit, had written ALneas : Cyneas of T hefdy : Pyrrhus Epirota:zx\A Alexander his forme: Clear ebus: Paufanias : Euangelus: Polybius fdimlitt frende to Scipio : Eupolemus: Iphicrates , Poffdonius: and very many other worthy Capitaines , Philofophers and Princes of Immortall fame and me¬ mory : Whole fayrefl floure of their garland ( in this feat) was ^Arithmetike ; and a litle perceiuerance,in Geometricall Figures . But in many other cafes doth ^Arith- metike (land the Capitaine in great ftede. As in proportionyng ofvittayles , for the Army, either remaining at a Ray : or fuddenly to be encrcafed with a certaine number ofSouldiers.-and for a certain tyme.Or by good Art to diminifh his com- pany,to make the victuals, longer to ferue the remanent, & for a certaine determi- ned tyme : if nede fo require. And fo in fundry' his other accounted, Recke- ninges,Meafurynges,and proportionynges,the wife, expert, and CircumfpeCt Ca¬ pitaine will affirme the Science of Arithmetike , to be one of his chief Counfaylors, direCtersand aiders* Which thing(by good meanes)was euident to the Noble, the Couragious , the Ioyall , and Curteous John , late Earle ofWarwickc. Who was a yong G entlernan , throughly knowne to very few . Albeit his lufty valiant- nes,force,and Skill in Chiualrous feates and exercifes:his humblenes,andfrende- lynes to all men, were thinges, openly , of the world perceiued. But what rotes (otherwife,)vertue had faftenedin his breft, what Rules of godly and honorable k.' hfe John Dee his Mathematical! P r^face . life he had framed to him felfe: what vices, (in fome then liuitig) notable, he tooke great care to efchew: what manly vertues , in other noble men , ( floriihing before his eyes,) he Sythingly afpired after : what prowefles he purpofed and menttoa*- chieue : with what feats and Artes,he began to furnifh and fraught him felfe , for the better feruice of his Kyng and Countrey sboth in peace & warrc. Thefe(I fay) his Heroicall Meditations , forecaftinges and determinations , no twayne , (I thinke )befide my felfe, can fo perfe&ly,and truely report. And therforedn Con¬ fidence,! count it my part, for the honor, preferment, & procuring of vertue (thus* briefly) to haue put his Name , in the Rcgifter of Fame ImmortalL T o our purpofe. This lohn, by one of his a&es (bdides many other .'both in En¬ gland and Fraunce,by me, in him noted. ) did difclofe his harty loue to vertuous Scien'ces:and his noble intent,to excell in Martiall proweffe: When he,with hum¬ ble requeft, and inflantSollicitingrgot the bed Rules (either in time paft by Greke or Romaine,or in our time vfed:and new Stratagemes therin deuifed) for ordring of all Companies/ummes and N umbers of me, (Many, or Few) with one kinde of weaponeer mo, appointed:with Artiliery,or without:on horfebacke, or on fote: to glue, or take onfet : to feem many, being few : to feem few , being many. T o marche in battaile orlornay : with many fuch feates,to Foughten field,Skarmoufb* or Ambufhe appartaining: And of all thefe,liuely deiignementes ( moft curioufly) to be i n velame parchement deferibed : with N otes & peculier markes,as the Arte T{. M requireth : and all thefe Rules.and deferiptions Arithmetical! , inclofedina riche Earlej'dyed Cafe of Gold , he vfed to weare about his necke : as his Iuell moft precious , and Anno. / y y Counfaylour moft trufty . Thus,o/r^«z^/^,ofhim,was ihryned in gold : Of fkar^e Numbers frute , he had good hope. N ow , N umbers therfore innumerable , in l!;”0!?? 'NumbersytzyfefviS fhryne ihall finde. fueHhis What nede I,(forfarder profe to you) of the Scholemafters of!uftice,to wife= Daugh" require teftimony :how nedefuil, how frutefull , how fkillfull a thing ^Arithmetike ^ is?I meane,the Lawyers ofall fortes. Vndoubtedly,the Ciuilians,can meruaylouf- merfet. ly declare ; how,neither the Auncien; Romaine lawes , without good knowledge of Numbers art ,cm be perceiued : Nor (Iuftice in Infinite Cafes) without due pro¬ portion, (narrowly confidered ,) is hable to be executed. How luftly, & with great knowledge of Arte, did PapirJanus inftitute a law of partition. , and allowance , be- twene man and wife after a diuorcec’But how Accurfms, Baldus,Bartalus,Iafon,Alex - under ^ and finally Alciatus, (being otherwife,notab!y Well learned)do iumble,geflTe, and erre,from the equity, art and Intent of the lawmaker : Arithmetike can dete<ft, and conuince: and clerely, make the truth to fhine. Good Bartolus , tyred in the examining & proportioning of the matterrand with Accurfms Gloffe, much cum- bredrburft o,ut,and faydiNu/la ejiin teto libro , hacglojfa diffcilior : Cuius computation nem nec Scholafiici nec Doctor es intclUgunt. &c . Thatis: Jn the Svhole booke , there is no Glojje harder then this : Whofe accoumpt or reckenyng 3 neither the Scho* lerspior the fdociours 'bnderftand.&c. What can they fay of lulianus law , Si ita Scriptum . efr . O f th e Teftators will iuftly performing, betwene the wife , Sonne and daughter < How can they perceiue the aquitie of Ayhricanus , Arithmetic all Reckening, where he treateth of LexFaicidiai How can they deliuer him, from his Reprouers : and their maintained : as Ioannes , Accurfms Hypclitus and alciatus? How luftly and artificially, was Africantis reckening madec'Proportionating to the Sommes bequeathed, the C on trib u tio ns ofeche part Namely, for the hundred prefently receiued,i7 ~ . And for the hundred, receiued after ten monethes,i2 A-; which make the 30: which were to be cotributed by the legataries to the heire» a.j. For, Inflict . ST j lohn Dee his Mathematical! Preface. For,what proportion^ oo hath to 75 : the fame hath 17 JL to 12 d_ : Which is Sef- quitertia: that is,as 4, to 3 .which id&key. Wonderfull many places, in theCiuile law, require an expert Arithmetics, to vnderftand the deepe Iudgemet,& Iuft de¬ terminate of the Auncient Romaine Lawmakers . But much more expert ought he to be, who iliouid be hable , to deride with tequitie, the infinite varietie of Cafes, which do, or may happen , vnder euery one of thole lawes and ordinances Guile. Hereby,eafely,ye may now conietiure: thatin the Canon law: and in the lawes of the Realme (which with vs , beare the chief Authoritte ) , Tuftice and e- quity might be greately preferred,and fkilfully executed, through due fkill of A- rithmetike,and proportions appertainy-ng. The worthy Philofbphers , and pru¬ dent Iawmakers(who haue written many bookes De Republica: How the belt ftate of Common wealthes might be procured and mainteined, ) haue very well deter¬ mined ofluftice : (which, not onely, is the Bale and foundation of Common weales :but alfo the totall perfection of all our workes, words, and thoughtes : jde- fining it, to be that vertue,by which, to euery onc,is rendred, that to him appertai¬ ned!. God challengeth this at our handes,to be honored as God: tobeloued,as a father : to be feared asaLord & mafter. O ur neighbours proportions1 alfo prcf- cribed of the Almighty lawmaker: which is > to do to other , euen as we would be done vnto. Thefe proportions, are in Iuflice neceflary :in duety, commendable: and of C ommon wealthes, the life,ftrength , flay and florilhing. ^AnjlotL in his Ethikes (to fitch the fede of Iuflice, and light of diretiion, to vfe and execute the fame) was fayne to fly to the perfection, and power of Numbers : for proportions Arithmetical! and Geometricall. Plato in his booke called Epinomis ( which boke, is the Threafury ofall his dotin' nc) where, his purpofe is, to feke a Science, which, when a man had it, perfectly :he might feme, and fo be, in d cde JYife . He, briefly, of other Sciences difcourfing,findeth them, not hable to bring it to paffe : Butofthe Science of Numbers, he fayrh. lUaypu numerum mortalium generi dedit,id profeflo ef fciet . Dxum atitem aliquem , magi* quamfortunam , ad falutem no fir am, hoc munus nobis arbitror contuhjfe . &c . Nam ipfum bonorum omnium Author em, cur non maximi boni, PrudcntiA dico , caufam arbitramuri T hat Science pioerely ,K>htch hath taught man* kynde number ,Jh all he able to bryng it to paffe. And }1 thinke}a certaine God , rather then fortune pto hauegiuen hs this gft, for our blijfe . Forptohy Jhould ~tye not hedge him/cho is the Author of all good things }to be alfo the caufe of the greatejlgo'o4t'bjng,name!yJVjfedome ? There, at length,he proueth Wifedome to be atteyned , by good Skill of Numbers . With which great T eftimony, and the manifold profes , and reafons , before exprefled , you may be fufficiently and fully perfiiaded : of the perfect Science of Arithmetike, to make this accounte ; That of all Sciences,next to Thsolcgie,it is mofl diuine,moft pure, mofl ampleand generall, moft profour.de , moft 1 ubtile ,moft commodious and mofl: neceflary . Whole nextSifter,is the Abfolute Science of Magnitudes: of which (by the Diretiion and aide of him, whole (jMagmtude is Infinite,and ofvs Incomprehenlible ) I now en- tend , fo to write , that both with the ^Multitude, and alfo with the c Magnitude of MeruaylouS and frutefull verities , you ( my frendes and Countreymen ) may be ftird vp, and awaked, to behold what certaine Artes and Sciences, (to our vn~ Ipeakabic behofe)our heauenly father, hath for vs prepared, and reuealed,by liin* dry Philo [others and c Mathematiciens . ffdth^Number and c_ Magnitude , haue a certaine Originall fede, ( as it were ) of an incredible property: and of man, neuer hable. Fully, to be declared . Of Number , an Y nit, and of ^Magnitude, a Poynte,doo feeme to be much like Origi- s Y nail / lohn Dee his Mathematicall Preface* nallcaufes : But the diuerfitie ncuerthelefle,is great . We defined an Vnit , to be a thing Mathematicall Indiuifible: A Point, like wife, we layd to be a Ma¬ thematical! thing Indiuifible. Andfarder , that a Point may haue a certaine de^ termined Situation: that is, that we may afligne,and prefcribe a Point, to be here, there , yonder. &c. Herein , (behold) our Vnit is free, and canabyde no bon¬ dage, or to be tyed to any place,or feat: diuifible or indiuifible . Agayne , by rea- fon,a Point may haue a Situation limited to him.- a certaine motion, therfore (to a place, and from a place) is to a Point inciden t and appertainyng. But an Vnit,c an not be imagined to haue any motion . A Point, by his motion, produceth , Ma¬ thematically^ line: (as we layd before)which is the firft kinde of Magnitudes,and mod fimple: An Vnit,cm not produce any number . A Line, though it be produ¬ ced of a Point moued,yet,it doth not confift of pointes : N umber , though it be not produced of an Vnit , yet doth it Confift of vnits , as a materiall caufe . But formally,N umber, is the Vnion, and VnitieofVnits . Which vnyting and knit- Numbed ting, is the workemanlhip of our minde: which,of diftinift and difcrete Vnits , ma- keth a N umber: by vniformitie,refulting of a certaine multitude of Vnits. And fo, euery number, may haue his leaft part,giuem- namely, an Vnit: But not of a Magni¬ tude, (no, not of a Lyne,)the leaft part can be giue:bycaufe,infinitly, diuifion ther- of,may be concerned. All Magnitude,is either a Line, a Plaine, or a Solid. Which Line, Plaine, or Solid, of no Senfe,can be perceiued, nor exactly by had (any way ) reprefented:nor ofNature produced: But, as ( by degrees ) Number did come to our perceiuerance: So,by vifible formes, we are holpen to imagine, what our Line Mathematicall, is. What our Point, is.So precile,.are our Magnitudes , that one Line is no broader then an other: for they haue no bredth : Nor our Plaines haue any thicknes.Nor yet our Bodies,any weight.-be they neuer fo large of dimenfio. Our Body es, we can haue Smaller, then either Arte or Nature can produce a- ny : and Greater alfo , then all the world can comprehend . Our leaft Mag¬ nitudes, can be diuided into fo many partes , as the greateft . As, a Line of an inch long, (with vs) may be diuided into as many partes, as may the diame¬ ter ofthe whole world , from Eaft to Weft .- or any way extended : What priui- ledges, aboue all manual Arte, and Natures might, haue our two Sciences Ma¬ thematically to exhibite,afid to deale' with thinges offuch power, liberty, fimplici- ty,puritie,and perfe&ionc' And in them,fo certainly,fo orderly ,fo ptecifeiy to pro- :cede:as,excellentis that workema Mechanical! Iudged , who nereftcanapproche to the reprelenting of workes, Mathematically demonftrated i And our two Sci¬ ences, remaining pure, and abfolute,in their proper termes,and in their owne Mat- ter: to haue, and allowe,onely fuch Demonftrations , as are plaine , certaine , vni- tierlall, and of an seternall Veritye'This Science of ^Magnitude, his properties,con- Geometric • ditions,and appertenances : commonly ,now is,and from the beginnyng , hath of * all Philofophers , ben called Geometric . But,veryly,with a name to bafe andfcant, for a Science of fuch dignitie and amplenes. And,perchaunce , that name, by co- mon and fecret confent,of all wifemen, hitherto hath ben fuffred to remayne.-that it might carry with it a perpetuall memorye, ofthe firft and notableft benefite, by that Science, to common people file wed : Which was , when Boundes and meres of land and ground were loft, and confoundedfas in %y/tf>yearely,with the ouer- flowyng of Nilas, the greateft and longeft riuer in the world ) or , that ground be¬ queathed, were to be a(figned:or, ground fold, were to be layd out : on (when dis¬ order preuailed)that Commos were diftributed into feueral ties. For, where, vpon thefe & fuch like occafios,Some.by ignorace, fome by negligece, Some by fraude, and fome by violence, did wrongfully limite,meafure, encroach, or challenge ( By a.ij. pretence lohn Dec his Mathematical! Preface. pretence ofiuft content, and meafure) thole Iandes and groundes : great lofle4dif- quictnes, murder, and warredidffull oft)enfue:Till,by Gods mercy,and mans In- duflrie,The perfect Science of Lines, Plaines, and Solides (like a diuine Iufticicr,) gaue vnto eiiery man, his owne. The people then,by this art pleaTured,and great¬ ly relieuedjin their Iandes iuft mearuring:& other Philofophers, writing Rules for land meafuring. betwene them both, thus, confirmed the name of Geometria, that is, (according to the very etimologie of the word)Land meafuring.Wherin,the peo¬ ple knew no farder, of Magnitudes vfe,but in Plaines: and thePbilofophers,ofthe, had no feet hearers, or Scholers.-farder to difclofe vnto , then of flat , plaine Geome¬ tric. And though, thefe Philofophers,knew offardervfe,and bed vnderflode the etymologye of the worde,yet this name Gcmetria, was of them applyed generally to all fortes of Magnitudes ; vnleaft, otherwhile, of Plato , and Pythagoras .* When KPlatt. 7. dt tliey would precifely declare their owne dodrine. Then, was * Geometria , with "Kip' xhemftudiumquod circa planum verfatur. But, well you may perceiue by Eudides Elementes , that more ample is our Science , then to meafure Plaines:and nothyng lefle therin is toughtf of purpolejthen how to meafure Land. An other namc,ther- fore,muft nedes be had, for our Mathematical! Science of Magnitudes : which re¬ garded! neither clod, nor turff: neither hill, nor dale. neither earth nor heauen; but is abfolute CM* egethologia .-not creping on ground , and daffeling the eye, with pole 55 perche,rod or lyne.-butliftyng the hart aboue the heauens,by inuifibie lines , and O* immortall beames meteth with the reflexions, of the light incomprehenfible: and r> fo procureth Ioye,and perfedion vnfpeakable. Of which true vfe of our c Me%e- thica,ov tJl-f egethologia, Diuine Plato feemed to haue good tafte,and iudgement.-and (by the name ol Geometric ) fo noted it -and warned his Scholers therof: as,in hys feuenth Dialog , of the Common wealth,may euidently be fene. Where (in La- tin)thus it is : right well mandated : P r of eclo, nobis hoe non hegabunt , Quicmfy vclpau. litlum quid Geometria gufiarunt, quin bac Scientia , contra, omnino fe habeat , quamde ea loquuntur , qui in ipfa verfantur . In Englifh, thus. Verely(f ay th Plato foohofoeuer bauef hut euen "Very litle flailed of Geometric, will not denye Tmto Vs , this : but that this Science ,is of an other condicion, quite contrary to thatftohich they that are exercifed in it , do fpeake of it. And there it followeth, of our Gecmetrie, gupd quaritur cognofcendi illtus gratia, quod femper eft, non dr eius quod oritur quandotf dr interit. Geometria, eius quod eft femper, Cognitio efl.^ttolletigitur{o Generofe vir) ad Veritatem^anmum-atfyita^ad Philofophandum preparabit cogitationemjvt adfopera con - uertamus -quajiuncyontra quam decetyid inferior a deijeimus. dre . Quam maximeigitur pracipiendum efL'vt qui praclarifsimam banc habitat Civitatem,nullo modo,Geometriam fernant . Nam dr quee prater ipfms propofitum,quodam modo effe videnturfhaud exigua font. drc.It inuft nedes be confdfed (faith Plato ) That £ Geometric} is learned , for the knowyng of that , mhich is euer.and not of that, *0 vhicb,in tymejboth is bred and is brought to an ende.iyc. Geometric is the knowledge of that which is euer * lajlyng. It mill lift Vp therfore( 0 Gentle Syr ) ourmynde to the Veritie : and by that meanest mill prepare the T bought, to the Thtlofophicall loue ofmifdome: that me may turne or conuert , toward heauenly t hinges iSett mjnde a*d thouShf\ mhich now ,otherwife then becommeth V>s,me call down on bafe or inferior things. <zsrc. Chiefly, therfore, Commaundement mufl be giuen , that ftich as do inhabit this mofl honorable Qtie,by no meanes, delpife Geometric. For euen thofe thinges &>»* tj itymhkhjn manner, feame to be , befide the purpofe of Geometric : are of m iohn Dee h is Adatfiematfcail Preface. no frriall importance . ^c. And befides the manifold vfes of Geometrie, in matters appertainyng to warre,he addeth more,offecond vnpurpofed frute, and commo* ditye,arrifing by Geometrie : fay in g : Seim us quin etiamyid Difciplinas omnes faciliusper difcendas ,1'Mereffe ommno,atpgerit ne G eometriam altquis,an non . &c. Hanc ergo D o- clrinam^fectmdo loco difeendam Imenibus Jlatmmus . That is. (But >alfo 3loe kno"%>} that for the more eafy learnyng ofallyfrtesft importeth much , whether one haue any knowledge in Geometrie }or no. <£rc. Let las therfore make an ordi* nance or decree , that this Science , of young men J. hall be learned in the fecond place. This was Diuine Plato his Iudgement,both of the purpofed , chief, and perfect vfe of Geometrie: and ofhis fecond, dependyng , deriuatiue commodities. And for vs,Chriften men, a thoufand thoufand mo occafions are, to haue nede of the helpe of* CMegethologicall Contemplations ; wherby,to tray ne our Imagina- * f . tions and Myndes ,by litle and litle,to forfake and abandon,the groffe and cortup- tible Obiedcs,of our vtward fenfes.-and to apprehend , by fure dodrine demon- ftratiue,Things Mathematical!. And by them , readily to be holpen and con- ' earthly name. t duCtcd to conceiue , difeourfe , and conclude of things Intellectual , Spiritual!, of Geometrie* £temall,and fuch as concerneour Bliffe euerlafting ; which, otherwise ( without Special! priuiledge of Illumination, or Reuelation fro heauen ) No mortall mans wyt( naturally) is hable to reach vnto,or to Compare. And,veryly,by my fmall T aient(from aboue)I am hable to proue and teftifie,that the litterall T ext,and or¬ der of our diuine Law,Oracles,ana Myfteries,require more fkill in Numbers, and Magnitudes .• then (commonly) the expofitors haue vttered : but rather onely (at the moftjfo warned : & (hewed their own want therin. (To name any, is nedeles: and to note the places, is, here, no place: But if I be duelyafked,my anfwere is rea¬ dy.) x^nd without the litterall,Grammaticall,Mathematicall or Naturali verities of fuch places , by good and certaine Arte,perceiued,no Spirituall fenfe ( propre to thofe places, by Abfolute T heologie) will thereon depend. N o man, therfore, can ^ doute 3 but toward the atteyning of knowledge incomparable , andHeauenly Wifedome.* Mathematicall Speculations, both ofNumbers and Magnitudes: are ” meanes, .aydes, and guides: ready, certaine , and neceflary. From henceforth,in this my Preface, will I frame my talke,to Plato his fugitiue Scholers: or, rather , to fuch, who well can,( and alfo wil,)vfe their vtward fenfes,to the glory of God,the benerite of their Coun trey, and their ownefecretcontentation, or honeft prefer¬ ment, on this earthly Scaffold. T o them,I will orderly recite, deferibe & declare a great Number of Artes , from our two Mathematicall fountaines , deriued into the fieldes of Nature. Wherby , fuch Sedes , and Rotes , as lye depe hyd in'the groud of ‘Nature, are refrefhed, quickened, and prouoked to grow, (bote vp, fioure, and giue frute, infinite,and incredible. And thefe Artes,fhalbe fuch , as vpon Mag¬ nitudes properties do depen de,more,then vpon N umber. And by good reafon we may call them Artes,and Artes Mathematicall Deriuatiue : for ( at this tyme)I *AnArtu Define An Arte, to be a Methodicall coplete Dodfrine, hailing abun- dancy of fufhcient,and pearlier matter to deale with, by the allow¬ ance of the Metaphificall Philofopher : the knowledge whereof, to humaine ftate is neceflarye. And that I account, An Art Mathemati- °*rt Math** call deriuatiue, which by Mathematicall demonftratiue Method, TuTtiue^ in Nubers , or Magnitudes, ordiah and confirmeth his dodtrine, as much & as perfedtly , as the matter fubiedt will admit . And for that, a.iij, I emend A Mechani- tietu I. Geometric vulgar . z. 1 . i. Note, Note, lohn Dee his Mathematical! Preface, I entend to vfe the name and propertie of a Mechanicien, o therwife,th en (hi th er to) it hath ben vfedj thinke it good, (for diftin&ion fake) to giue you alfo a brief det cription, what I meane therby. A Mechanicien,or a Mechanicali work¬ man is he , whofe f kill is , without knowledge of Mathematical! demonftration , perfectly to worke and finifhe any fenfible worke, by thd Mathematicien principall or deriuatiue, demonflrated or de- monilrabie. Full well I know, that he which inucnteth, or maketh thefe de- monftrations,is generally called ffeculatiue CMechanicien : which differreth no- thyng from a Mechanicali 'jMatkematicicn . So, in refpcd of diuerfe adtions,one man may haue the name offundry artes:as,fome tyme,ofa Logicien , fome tymes (in the fame matter otherwife handled) of a Rethoricien . Of thefe trifles,I make, (asnow,in refped of my Preface, )fmall account: to fyle the for the fine handlyng offubtile curious difputers . In other places , they may commaunde me, to giue good reafon : and yet, here, I will not be vnreafonable. Firft, then, from the puritie,abfolutenes,and Immaterialitie of Principall Geo¬ metric, is that kindc of Geometric deriued , which vulgarly is counted Geometric : and is the Arte of Meafuring fenfible magnitudes, their i'ufft quatities and contentes . This, teacheth to meafure,either at hand: and the praftifer, to be by the thing Meafured.* and fo,by due applying of Cumpafe, Rule, Squire, Yarde,Ell,Perch,Pole,Line,Gagingrod,(or filch like inftrument) to the Length, Plaine,or Solide meafured, '‘to be certified, either of the length, perimetry, or di- fiance lineall : and this is called, UMecometrie . Or* to be certified of the content of any plaine Superficies : whether it be in ground Surueyed, Borde, or Glafle mea- fiired,or fuch like thing : which meafuring,is named Embadometrie . *Or els to vn- derfland the Soliditie,and content of any bodily thing : as ofTymber and Stone, or the content ofPits,Pondes,Wells,Veffels,fmaU& great,of all fafhions.Where, ofWine,Oyle,Beere,or Ale veffells,&c,the Meafuring-commanly, hath a pecu- lier name.-and is called Gaging . And the generallname ofthefe Solide meafures, is Stereometric . Or els, this vulgar Geometric , hath confideration to teach the prac- tifer , how to meafure things, with good diflance betwene him and the thing mea¬ fured : and to vnderfland thereby,either *how Farre,athingfeene(on land or wa¬ ter) is from the meafurer: and this may be called Jfornecometrie: Or,how High or depe,aboue or vnder theieuel of the meafurers ftading,any thing is, which is fene on land or water, called Hypfometrie.*Qi it informeth the meafurer , how Broad any thing is, which is in the meafurers vew:fo‘itbc on Land or Water,fituated:and may be called Plat ometrie . Though I vfe here to condition,the thing meafured, to. be on Land, or Water Situated : yet, know for ceitaine, that the fundry heigthe of Cloudes, blafing Starres, and of the Mone ,may(by thefe meanes)haue their di- flances from the earth : and, of the blafing Starresand Mone,the Soliditie (afwell as difiances) to be meafured:But becaufe, neither thefe things are vulgarly taught: nor of a common praftifer fo ready to be executed V I,rather,let fuch meafures be reckened incident to fome of our other Artes, dealing with thinges on high,more purpofely, then this vulgar Land meafuring Geometrie doth : as in Perjpetfiue and t^AUronomie, &c . f QF . thefe feates ( Farther applied ) is Sprongthe feateof Geodejie , or Land Meafuring: more cunningly to meafure & Suruey Land, Woods, and Waters, a farre of. More cunningly, I fay :' But God knoweth (hitherto) in thefe Realmes of England and Ireland ( whether through ignorance or fraude , I can not tell , in e,uery particular ) how great wrong and iniurie hath (in my timejbene committed lohn Dee His Mathematical! Preface, by vntrue meafuring and furueying ofLand or Woods, any way . And, this I art! fure: that die Value of the difference, bet wene the truth and fuch Suru eyes, would haue bene hable to haue loud (for euer) in eche of our wo Vniuerfitics,an excel¬ lent Mathematicall Reader: to eche,allowing (yearly) a hundred Markes oflawfull money of this realme: which, in dede,would feme requifit,here,to be had (though by other wayes prouided for) as well,as,the famous Vniuerfitie of Paris, hath two Mathematicall Readers : and eche, two hundreth French Crownes yearly, of the French Kinges magnificent liberalitie onely » Now,againe, to our purpofe retur¬ ning : Moreouer, of the former knowledge Geometricall,aregrowen the Skills of Geographic , Chorographie , Hydrographic , and Stratarithmetrie . Geographic teacheth wayes, by which, in fudry formes, (as Sph<zrike,Vlaine n or other) ,the Situation of Cities, Townes,Villages, Fortes,CaftelIs,Mountaines, „ Woods,Hauens,Riuers,Crekes,& fuch other things,vpo the outface of the earth- „ ly Globe (either in the whole,or in lome principall meter and portion therofco- „ tayned)may be defcribed anddefigned, in comenlurations Analogicall to Nature and veritierand moft aptly to our vew,may be reprefented.Of this Arte how great 7f pleafure,and how manifolde commodities do come vnto vs,daily and hourely : of moft men, is perceaued . While, fome, to beatitifie their Halls,Parlers, Chambers, Galeries,Studies,or Libraries with: other fome,for thinges paft, as battels fought, earthquakes, heauenly fyringes,&fuch occurentes,in hiftories mentioned: therby liuely ,as it were, to vew e the place,the region adioyning,the diftance from vs : and fuch other circumftances . Some other, prefently to vewe the large dominion of theTurke : the wide Empire of the Mofchouite: and thelitle morfell of ground, where Chriftendome(by profeffion)is certainly knowcn. Litle,Ifay,in rclpe&e of the reft, &c. Some, either for their owne iorneyes direding into farre landcs: or to vnderftand of other mens trauailes . To conclude, fome, for one purpofe ; and fome, for an other, liketh,loueth,getteth,and vfeth, Mappes, Chartes,& Geo* graphical! Globes . Ofwhofe vie, to fpeake fuff ciently, would require a booke peculier. Chorographie feemeth to be art vnderling, and a twig, of Geographic: and y et neuerthelefte, is in pradile manifolde, and in vfe very ample . This tea- ,5 cheth Analogically to defcribe a fmall portion or circuite of ground, with the con- „ rentes : not regarding what commenfuration it hath to the wholes, or any parcell, „ without it, contained . Butin the territory or parcell of ground which it taketh in » hand to make defcription of, itleaueth out (orvndefcnbed) no notable , or odde „ thing, aboue the ground vifible .Yea and fometimes , of thinges vnder ground, ,, geueth fome peculier marke .* or warning : as ofMettall mines, Cole pittes, Stone „ quarries. &c. Thus, a Dukedome,a Shiere,a Lordfliip, or Idle, may be defcribed „ diftindly . But marueilous pleafant, and profitable it is , in the exhibiting to our eye, and comhienfuration, the plat of a Citie, Towne, Forte, or Pallace, in true Symmetry : notapproching to any of them : and out of Gunne fhot.&c. Hereby, the K^frchitett may furnifhe him felfe, with ftore of what patterns he liketh : to his great inftrudion: euen in thofe thinges which outwardly are proportioned: either limply in them felues : or refpediuely,to Hilles,Riuers, Hauens, and Woods ad- ioyning . Some alfo, terme this particular defcription of places , Topographic . HydrOgraphlC,deliuereth to our knowledge , on Globe or inPlaine, „ the peifed Analogicall defcription of the Ocean Sea coaftes, through the whole world : or in the chiefe and principall partes thereof : with the lies and chiefe sj adfij* paticular *Nate', Thedijfe- ,, rence be- „ ttyene Stra- „ tarithme - ,, trie and. 3J Tafiicie, lohn Dee his Mathematical! Preface. particular places ofdaungers, conteyned within the boundes.,and Sea coafteS de- icribed : as, of Quicldandes,Bankes-,Pittes,Rockes,Races,CountertideSiWhorle» pooles. &C. This, dealeth with the Element of the water chiefly ; as Geographic, did principally take the Element of the Earthes defcription ( with his apperte- nances ) to taske . And befides thys , Hyd.rographie , requireth a particular Regifter of certaine Landmarkes (where markes may be had) from the fea,well lia¬ ble to be fkried, in what point of the Seacumpafe they appeare,and what apparent form e,S.ituation, and bignes they haue, in refpe&e of any daungerous place in the fea,or nere vnto it, affigned: And in all Coaftes, what Morte,maketh full Sea.-and what way, the Tides and Ebbes, come and go, the Hydrographer oughtto recorde. The Saundinges likewife : and the Chanels wayes: their number, and depthes or¬ dinarily, at ebbe and flud, ought the Hydrographer , by obferuation and diligence of Measuring, to haue certainly knowen . And many other pointes,are belonging to perfede Hydrographies and for to make a Rutter, by : of which,I nede not here fpeake : as of the defcribing,in any place, vpon Globe or Plaine, the ^a.poin’tes of the Compafe,truely : (wherof, fcarflyfoure, in England , haue right knowledge: bycaufe, the lines thcrof, are no (iraight lines , nor Circles . ) Of making due pro- iedion of a Sphere in plaine.Of the Variacion of the Compas , from true N orthe: And fuch like matters (of great importance , all ) I leaue to fpeake of in this place: bycaufe, I may feame(al ready)to haue enlarged the boundes,and duety of an Hy- dographer, much more,then any man (to this day)hath noted, or preferibed . Yet am I well hable to proue,all thefe thinges , to appertaine , and alfo to be proper to the Hydrographer. The chief vfe and ende of this Art, is the Art ofNauigation: butit hath other diuerfe vies : euen by them to be enioyed , that neuerlacke fight of land. Stratanthmetrie, is the Skill, (appertainyng to the warre , ) by which a .man can fee in figure,analogicall to any Geometrical figure appointed, any certaine number orfumme of men: offuch a figure capable: (by reafon ofthe vfuall (paces betwene Souldiers allowed : and for that , of men,can be made no Fradions. Yet, neuertheles,he can order the giuen fumme of men , for the greateftfuch figure, that of them, cabe ordred)and certifie,of the ouerplus: (if any be) and of the next certaine fumme, which, with the ouerplus,will admitafigureexadly proportionail to the figure affigned. By which Skill,alfo,ofany army or company of men : (the figure & fides of whole orderly (landing, or array, is knowen)he is able tp exprefle the iufl number of men, within that figure coriteined: or(orderly ) able to be con- teined. * And this figure, and (ides therof, he is hable to know : either beyngby, and at hand: or a farre of. Thus farre, ftretcheth the defcription and property of Stratarithmetrie : furficienr for this ty me and place . It difterreth from the Feate Ta£licall,De dciebus injlruendd. bycaufe, there, is neceffary the wifedome and fore¬ fight, to what purpofe he fo ordreth the men ; and Skillfull liability , alfo , for any occafion,or purpofe , to deuife and v(e the apteft and mod neceflary order', array and figure of his Company and Summe of men . By figured meane.- as,either ofa Perfect Square, Triangle , Circle , Ouale , longfquarc , (of die Grekes it is called Eiero - rmkes ) Rhombe, Rhomboid, Lunular , Ryng, Serpentine, and fuch other Geometricall figures: Which,inwarrcs, haue ben, and are to be vfed : for commodioufhes , ne- ceflity,and auauntage &c. And no (mail f kill ought he to haue , that lhould make true reporter nere the truth,of the numbers and Summes,offootemen or horfe- men , in the Enemyes ordring . A farre of, to make an eftimate , betwene nere termes of More and Leffe,is not a thyng very rife , among thofe that gladly would John Dee his Mathematical Preface. do it. Great pollicy may be vfed of the Capitaines,(ar tymes fete, and in places t.fil conuenientjas to vfe Figures , which make greateft fhew , offo many as hehath: ou ^fn‘ie an d vfing the aduauntage of the three kindes of vfuall fpaces r ( betwene footemen it hard, to perforin or horfemen)to take the largefbor when he would feme to haue few, (beyng ma- ny.- ) contrary wife, in Figure, and Ipace. The Herald, Purfeuant, Sergeant Royally ^Zheyl^ifi Capitaine , or who foeuer is carefull to come nere the truth herein , befides the Iudgement ofhis expert eye,his fkill of Ordering TacUcall, the helpe of his Geo- Sides {ink Angles J metricall inuru men t : Ring, or Staffe Aftronomicall : ( commodioufly framed for •And where. Refold cariage and vfe) He may wonderfully helpe him ftlfe* byperfpediue Glaffts.In which, (I truft ) our pofterity will proue more fkillfull and expert , and to greater ^ITg^utdaL. purpofes, then in thefe dayes, can (almoft )be credited to be poflible. generally with *A- Thus haue I lightly pafied otter the Artificiall Feates,chiefly dependyng Vpoil and.thatfor Bat - vulgar Geometric : & commonly and generally reckencd vnder the name of Geome- trie. But there are other(very many) AMethodicall Artes j which, declyning from the purity , •implicitie,and Immateriality, of our Principall Science of Magnitudes: CA ' do yet neuertheles vfe the great ayde , direction , and Method of the fayd principall Science , and haue propre names , and diftind : both from the Science of Geometric, (from which they are deriued)and one from the other. As Per- fpeediue, Aftronomie , Muhke, Cofmographie, Aftrologie,Statike, Antbropographie^rochilike;, Helicofophie, Pneumatithmie, Me- nadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Archi¬ tecture, Nauigation , Thaumaturgike and Archemaftrie. I thinke it neceflaiy , orderly , of thefe to giue fome peculier deferiptions : andwithall, to touch fome of their commodious vfes , and fo to make this Preface , to be a litde fwete,pleafant N ofegaye for you.-to comfort your Spirites , beyng almoft out of courage, andindefpayre, ( through brutifh brute ) Weenyng that Geometric ,had but ferued for buildyng of an houfe,or a curious bridge, or the roufe of Weftmin- fter hall , or fome witty pretty deuife , dr engyn , appropriate to a Carpenter,or a loyner &c.That the thing is farre otherwife , then the world , (commonly) to this day, hath demed,by worde and worke , good profe wilbe made. Among theft Artes, by good reafon,P erlpectiuc oughtto be had , ere of l. Aftronomicall Apparences , perfed knowledge can be atteyned. And bycaufe of the prerogatiue of Light , beyng the firftof Gods Creatures: and the eye, the light of our body, and hisSenfe moft mighty, and his organ moft Artificiall and Geome- tricall: At Pe'rpffiue, we will begyn therfore. Perfpediue,is an Art Mathe¬ matical!, which demonftrateth the maner^and properties, of all Ra¬ diations DireCt, Broken, and Reflected. This Defcription,or Notation , is briefbut it reacheth fo farre,as the world is wyde. It concerneth all Creatures, all Adions , and palfions, by Emanation of bearnes perfourmed . Beames,or na- turalllines , (here) I meane , notoflight onely,or of colour (though they,to eye, giue fhew, witnes, and profe , wherby to ground the Arte vpon )but alfo of other F ormes, both Subjlantiall, and Accidentally the certain e and determined adiue Ra« diall emanations. By this Art(omitting to fpeake ofthehigheft pointes) we may vfe our eyes, and the light,with greater pleafure:and perfeder Iudgementrboth of things, in light feen,& of other: which by like order of Lightes Radiations, worke and produce their effedes . We may be afhamed to be ignorant of the caufe,why fo fundry wayes our eye is deceiued,and abufedras, while the eye weeneth a roud Globe or Sphere(beyng farre of) to be a flat and plain e Circle,and fo likewife iud- b.j. geth <sV*T* ■.iWwr •v.t, -- A maruciloHs Clap. (TJ3 s.mp. John Dee his Mathematical! Preface. geth a plaine Square, to be roud : fuppofeth walles parallels,to approche,a farre of*: rofe and floure parallels,the one to bend downward , the other to rife vpward,at a little diftance from you. Againe , of thinges being in like fwiftnes of mouing , to thinke the nerer,ta moue faftenand the farder,much flower.Nay, of two thinges, wherof the one (incomparably) doth moue fwifter then the other , to deme the flower to moue very fwift,& the other to ftand: what an error is this,ofour eye? Or the Raynbow, both of his Colours, of the order of the colours, of the bignes of it, the place and heith ofit,(&c)to know the caufes demonflratiue,is it not pleafant, is it not necdlaryc'of two or three Sonnes appearing: ofBlafing Sterres : and fuch like thinges : by naturall caufes , brought to paffe , (and yet ncuertheles , offarder matter, Significatiue ) is it not commodious for man to know the veiy true caufe, & occafion Natural! i Yea,rather,is it not, greatly, againft the Souerainty of Mans nature , to be fo ouerfhot and abufed , with thinges ( at hand ) before his eyes { as with a Pecockes tayle , and a Doues necke : or a whole ore, in water, hob den, to feme broken . * Thynges, farre of, to feeme nere : and nere, to feme farre of . Small thinges , to feme great : and great , to feme final! . One man, to feme an Army . Or a man to be curftly affrayed of his owne (had- do w . Yea ,fo much, to feare,that,if you,being(alone ) nere a certaine glaffe , and proifer,with dagger or fword,to foyne at the glafle , you fhall fuddenly be irioued to gitiebacke(inmaner) by reafon of an Image, appearing in the ayre,betwene you & the glafle, with like hand, /word or dagger,& with like quicknes , foyningat your very eye, likewife as you do at the Glafle. Straunge,this is, to heare of: but more meruailous to behold, then thefe my wordes can fignifie. And neuerthe- lefle by demonfiration Opticall, the order and caufe therof, is certified: euen fo,as the effe<fl is confequent. Y ea,thus much more, dare I take vpon me, toward the fa- tiilying of the noble courrage, that longeth ardently for the wifedome of Caufes Naturalhas to let him vnderfiand, that, in London , he may with his owne eyes, haue profe of that, which I haue layd herein . A Gen deman, (which,for his good feruice, doneto his Countrey,is famous and honorable : andforfkill in the Ma¬ thematical! Sciences, and Languages, is the Od man of this land. &c. ) euen he,is hable:and(I am furc)will, very willingly, let the Glafle, and profe be fene.-andfo I (here) requeft him : for the encreafe of wifedome , in the honorable : and for the flopping of the mouthes malicious : and reprdfing the arrogancy of the ignorant. Y e may eafily gefle , what I mcane. This Art of Perjpmm r, is of that excellency, and may be led, to the certifying, and executingoffuch thinges , as no man would eafily beleue: without Adtuall profe perceiued. I fpeake nothing of Naturall Phi- tefopbie, which,' without Perjj’cctive, can not be fully vnderftanded , nor perfectly at- teined vnto. N or, of Jjlronomie: which, without P erjpecHue ^czn not well be groun¬ ded : N or '^Afirdogic , naturally Verified, and auouched. That part hereof,which dealeth with Glaflcs(which name,Glafle,is a generall name, in this Arte , for any thing, from which, a Beame reboundeth) is called Catoptrike > and hath fo many v- fes,both merueiloiis,and proffitable: that, both, it would hold me to long , to note therin the principall conclufions,all ready knowne: And alfo(perchaunce) fome thinges,mightlackedue credite with you : And I, therby, to leefe my labor:and g^. you, to flip into light Iudgement*, Before you haue learned fufficiently thepowre of N ature and Arte. IMow , to procede: «AftronoiTllC,is an Arte Mathematical!, which demonftrateth the difbmce , magnitudes , and all naturall motions, apparences,and pafsions propre to the Planets and fixed Sterres : for " 1 any IohnDee his Mathematical Preface** ^any time paft, prefen t and to come:in relpecf of a certaine Horizon i or without reipect of any Horizon. By this Arte we are certified of the di- ftance of the Starry Skye,and of eche Planete from the Centre of the Earth.- and of the greatnes of any Fixed ftarre fene, or Planete, i n reiped of the Earthes greatnes. As , we are lure ( by this Arte ) that the Solidity , Mailines and Body of the Sonne, conteineth the quantitieofthe whole Earth arid Sea, a hundred threfcoreand two times , leffe by ~ one eight parte of the earth. But the Body of the whole earthly globe and Sea, is bigger then the body of the Mone , three and forty times lefie by -- of the Mone. Wherfore the Sonne is bigger then the CVtone , 7000 times, idle, by 55? C- that is , precifely 69 40 11 bigger then the CMone. And yet the vnfkillfuli man, Would iudge them a like bigge . Wherfore,of Necefsity,the one is much farder from vs, then the other. The Sonne , when he is fardeft from the earth (which, now, in our age, is, when he is in the 8 .degree,of Cancer)is , 1179 Semidiameters of the Earth, diftante . And the c M one when file is fardeft from the earth, is 68 Semidiameters ofthe earth and — The nereft , that the CMone com- meth to the earth, is Semidiameters 52 ~~ The didance ofthe Starry Skye is ,fro vs, in Semidiameters of the earth 20081 ~C' Twenty thoufand fourefcore , one, andalmoftahaifc. Subtract from this, the CM ones nereft diftance,from the Earth: and therof remaineth Semidiameters of the earth 200.29 _l Twenty thoufand nine and twenty and a quarter. So thicke is the heauenly Palace , that the Pla~ netes haue all their exercife in, and moft meruailoufly perfourme the Commaude- inent and Charge to them giuen by the omnipotent Maieftie of the king of kings. This is that, which in Genefis is called Ha Rakia . Confident well. The Semidia- meterof the earth, coteineth of our common miles 3436-1. three thoufand, foure hundred thirty fix and foure eleuenth partes of one myle.-Such as the whole earth and Sea, round about, is 21600. Oneand twenty thoufand fix hundred of our myles. Allowyng for euery degree of thegreateft circle, thre fcore myles. Now if you way well with your felfe butthislitleparcell offrute ^flronomcall, as con¬ cerning the bignefte,Diftances of Sonne, Mone, Sterry Sky, and the huge maffinesof Ha Rakia , will you not finde your Confciences moued , with the kingly Prophet, to fingthe confeffion of Gods Glory, and fay, TheHeauens declare the gW* ry ofGodydncl the Firmament R«ki«\ Jbeweth forth the 1 vork.es of his handes. And fo forth, for thofe fiue firft ftaues,of that kingly Pfalme. Well,well,It is time for fome to lay hold on wifedome, and to Iudge truly of thinges: and notfo to ex¬ pound the Holy word,all by Allegories : as to Negleft the wifedome, powre and Goodnes ofGodpn, and by his Creatures , and Creation to be feen andleamed. By parables and Analogies of whofe natures and properties,the courfe ofthe Ho¬ ly Scripture, alfo, declareth to vs very many Myfteries.The whole Frame of Gods Creatures, (which is the whole world, )is to vs, a bright glafie: from which, by re¬ flexion, reboundeth to our knowledge and perceiuerance, Beames , and Radiati¬ on s.-reprefenting the Image of his Infinite goodnes, Omnipotecy,and wifedome. And wc therby , are taughtand perfuaded to Glorifie our Creator,as God.-and be thankefull therfore . Could the Heatheniftes finde thefe vfes,of thefe moft pure, beawtifulLand Mighty Corporall Creaturesrand fliall vve, after that thetrue Sonne ofrightwifenefie is rifen aboue the Horizon, of our temporall Hemijjharieyiod. hath fo abundantly ftreamed into our hartes,the direft: beames ofhis goodnes , mercy, and grace: Whofe heat All Creatures feele : Spirituall and Corpora)].- Vifible and b.ij. Inui- lohn D ee his Mathematical! Preface » Inuifible.-S hall we (I fay)looke vpon the Heauen, Stems, and Planets,^ an Oxe and an AlTe doth: no furder carefull or inquifitiue, what they are.- why were they Cre¬ ated, How do they execute that they were Created forc'SeingJAll Creatures,were for our lake created : and both we, and they, Created, chiefly toglorifie the Al¬ mighty Creator: and that, by all meanes,to vs poffible. Noliteignorare{i aith Plate in Epnomis ) Aftronomiam, Sapientiftimu quiddam ejfe. Be ye not ignorant jjfJlro* notnie to be a thyng of excellent yoifedome. cpflronomie, wzs to vs,from the be¬ ginning commended, and in maner commaunded by God him felfe.In afinuch as he made the Sonne, CMone, and Stems, to be to vs, for Sijgnes,an& knowledge ofSea- fons,and for Diftin&ions oFDayes,and yeares, Many wordes nede not . But I wifh,euery man fliould way this wor d,Signes. And befides that, conferre it alfo with the tenth Chapter of Hteremie. And though Some thinke , that there, they haue found a rod.- Yet Modeft Reafon,wiIl be indifferent Iudge, who ought to be beaten therwith,in relped of our purpofe. Leauing that : I pray you vnderftand this : that without great diligence of Obferuation , examination and Calculation, their periods and oourfes(wherby Dift'intfion ofSeafons,yeares,and New Mones might precifely be knowne) could not exa&ely be certified . Which thing to per¬ formers that Art , which we here haue Defined to be Aftronomie. Wherby , we may haue the diffindl Courfe of Times, dayes, yeares, and Ages: afwell for Confi- deratio of Sacred Prophefies,accomplifhed in due time, foretold ; as for high My- fticall Solemnities holding.- And for all other humaine affaires , Conditions , and couenanres , vpon certaine time ,betwcne man and man • with many other great vfcs.- Wherin , ( verely) , would be great incertainty, Confufion,vn truth, and bru- tifli Barbaroufnes: without the wonderfull diligence and fkill of this Arte : conti¬ nually learning, and determining Times, and periodes of Time , by the Record of the heauenlybooke , wherin all times are written .- and to be read with an Aftrono - ■me all fidffe, in ftede of afcftue. Muflke , of Motion, hath his Original! caufe .- Therfore , after the motions moft fwift,and moft Slow, which are in the Firmament, of Nature perfonned:and vnderthe Astronomers Conftderation -now I will Speake ofan other kinde of Motion , producing fou nd,audible,and of Man numerable. CMufikel call here that Science, Which of the Grekes is called Harmonic* . Not medling with the Controueriie be- tweiie the auncient Harmoniftes,2.rtd Canoniftes . Nf ufllce is a Ninth ernaticaJJ. Science , which teaebtetb,by fenfe and reafon, perfectly to iudge, and oTcfef the diiierfities of foundes,hye and low. Aftronomie and cMuftke are Sifters, faith Plato. As, for Aftronomie, the eyes : So, for Harmonious Motion, the eares were made. But as Aftronomie hath a more diuine Contemplation , and co- modity,then mortall eye can perceiue : So, is c Muftke to be confidered,that the i , * Minde may be preferred,before the care . And from audible found, we ought to afeende , to the examination : which numbers are Harmonious which not. And why, either, the fee arc : of the other are not. I could atiarge,in theheauenly 2. * motions and diftances , defcribe ameruaiIous Harmonie , o£ Pythagoras Harpe ^ . with eight ftringes. Aifo,fbmwhat might be fay d of Mercurius* two Harpes, eche of foure Stringes Elementall. And very ftraunge matter, might be alledged 5 . of the Harmonie, to our * Spiritual! part appropriate. As in Ptolomms rhird boke, in the fourth andfixth Chapters may appeare . * And what is the caufe of the apt 6 bondtymd frendly felowlhip,of the Intelle&uall and Mentall part of vs , with our groftc& corruptible body : but a certaine Meane, and Harmonious Spirituality, with both lohn Dee his Mathematical! Preface. both participatyng^ of both (in a maner)reftdtyngi In the * Tune of Mans <voyce,and alp * the found of Infrument^dmx might be fay d, of Harmonic: N o common Muficien g . would lightly beleue.But of the fundry Mixture(as I may terme it) and concurfe, I'D* diuerfe collation, and Application of th efe Harmonies: as of thre,foure,fiue,or mo: Read in A* Maruailous haue the efre&es ben: and yet may be foundc,and produced the like: r^°dls!ns - with feme proportional! confidcration for our time, and being : inrdpedt of the State , of the thinges then : in which , and by which , the wondrous effe&es were tH^'an£ wrought. Democritus and T heophrasius affirmed, that, by CELufike, griefes and di- y. chapters, feafes of the Minde,and body might be cured, or inferred. And we finde in Re- where you cord e, that T er pander, Anonffmemas^ Orpheus ^Amphion,Dauid, Pythagoras^ Empedo- Jkall bane cles, jjclepiadesymd T imotheus, by Harmonicall Confonacy,haue done, and brought fom£ occafion to pas, thinges, more then meruailous , to here of. Of them then, making no far- ffder to der difcourfe,in this place : Sure I am, that Common Mufike , commonly vfed, is found to the c Muficiens and Hearers,to be fo Commodious and pleafant , That if commonly is I would fay and difpute,but thus much: That it were to be otherwife vfed , then it thought. is, I iliould finde more repreeuers, then I could finde priuy,or fkilfull of my mea¬ ning. In thinges therfore euident,and better knowen,then I can exprefTe:andfo allowed and liked of, (as I would wilh/ome other thinges, had the like hap) I will {pare to enlarge my lines any farder,but confequently follow my purpofe. Of Cofmograpllie,! appointed briefly in this place, to gcue youforae intelligence. Cofmographie,is the whole and perfed defeription of the heauenly,and alfo elementall parte of the world , and their ho* mologall application , and mutuall collation necelfarie. This Art, requireth AHronomie , Geographic , Hydrographie and CMttfike . Therfore ,it is no finall Arte, nor fo fimple,as in common pra&ife, itis(iiightly)confidered. This marcheth Heauen, and the Earth,in one frame,and aptly applieth parts Correfpo- dentrSo ,as, the Heauenly Globe, may (in pradife) be ducly deferibed vpon die Geographicall , and Hydrographical! Globe . And there , for vs to confider an Aqutnociiall Circle , an Ecliptike line , Colures, Poles, Stems in thei r true Longitudes, Latitudes,Declinations,and Verricajitie.-alfb Climes, and Parallels:and by an Ho¬ rizon annexed, and reuolution of the earthly Globe(as the Heauen,is, by the Pri- fnduantfi tied about in 24.requall Houres) to learne the Rifinges and Settinges of S terres (of Virgill in his Georgikes: of Hefod: of Hippocrates in his Medicinall Sphere, to Perdicca King of the Macedonians: of Diodes pto King Antigonus , and of other fa¬ mous P hilo fop hers prefcriBecra thing necefTary,for due manuring of the earth , for Nauigation, for the Alteration ofmans body:being,whole,Sicke,wounded,or bru- fed. By the Reuolution , alfo , or mouing of the Globe Cofmographicall , the Riling and Setting of the Sonne: thcLengthcs,ofdaycs and nightes : the Houres and times (both night and dayjareknowne : with very many other pleafant and neceffaty vfes : Wherof, fome are known chut better remaine, for fuch to know and vfe. who of a fparke of true fire,can make a wonderfull bonfire, by applying of *■>« due matter, duely. 1 rr 3 ° Of Aftrologie , here I make an Arte, feuerall from AHronomie : not by new deuife, but by good reafon and authoritie : for, Aftrologie, is an Arte Mathematical! , which reafonably demonftrateth the operations and effedles, of the naturall beames, of light, and fecrete influence: of the Sterres and P lanets. : in euery element and elementall body: b.iih at lohn Dee Lis Mathem aticall P raeface , at all times , in any Horizon affigned . This Arte is furniihed with ma¬ ny other great Artes and experiences: As with perfe&e Perftecliue, ^JHronomic, Cofmographie, Natural! Philofophie of the /{..Elementes, the Arte of Graduation, and fome good vnderftading in CMuftke : and yet moreouer, with an other great Arte, hereafter following, though I, here, fet this before, for fome confiderations me’ mouing. Sufficient (you fee) is the ftuffe, to make this rare and fecrete Arte, of: and hard enough to frame to the Conclufion Syllogifticall . Yet both the mani- fblde and continuall trauailes of. the raoft auncient and wife Philofophers,for the atteyning of this Arte : and by examples of effedes , to confirme the fame ; hath left vnto vs fufficient proufe and witndfe • and we,alfo,daily may pcrceaue , That mans body, and all other Elementall bodies, are altered, difpofed, ordred, pleafu- red, and difpleafured, by the Influential! working of the Sunne,Monc, and the other Starres and Planets . And therfore,fayth AriBotle^ in the firft of his Meteorological!. bookes, in thefecond Chapter Eft antem necejfario Mundus iUefitpernis lationibus fere contimus ; V t, mde, vis tins vniuerfta regatur . Ea ftquidcm Caufa puma pit and a omnibus eft, vndemotus principium exisnt. That is: This i Elementally World is of necefiitie, almojl , next adioyning/o the heauenly motions : T hat from thence , nil his loertne or force may hegouerned. For /hat is to he thought thefirf Caufe 'Pntoall ; from y>hich,the beginning of motion, is . And againe, in the tenth Chapter, Opcrtet igitur & bomm principia fumamns , (ft cattfits omnium ftmiliter. Principium igitur vt mouenspracipuumcf (ft omnium primum , Cir cuius tile eftjn quo manifefte S oils latio , &c . Andfo forth . His Meteor ologicall bookes, are full ofargu- mentes,and effeduall demonftrations,ofthe vertue, operation, and power of the heauenly bodies, in and vpon the fower Elementes, and other bodies, of them (either perfectly, or vnperfedly ) compofed, Arid in his fecond booke. Be Genera- tione drCorruptione , in the tenth Chapter , flup circa eftprima latio , Ortusfr Interi- tus caufa non eft: S ed obliqui Circuit latio : ea namift (ft continua ejlftft daobus motibus ft: In Erigliffie, thus . Wherefore the uppermost motion, is not the caufe of Gene * ration and Corruption, but the motion of the Zodiake : for , that , both, is con * tinuall , and is caufed of two mouinges . And in his fecond booke, and fecond Chapter of hys Phyfikes. Homo nam^generat hominem, at % Sol. For Man (lay th he) and the Sonne, are caufe of mans generation . Authorities may be brought, very many : both of 1000.2000.yea and 3000. y.eares Antiquitie : of great Philo- fophers. Expert, Wife, arid godly men,for that Conclufion: which,daily and houre- dy,we men,may difeerne and perceaue by fenfe andreafon : All beaftes do feele, andfimplyihew, by their actions and paffions, outward and inward ; All Plants, Herbes, Trees, Flowers, and Fruites . And finally, the EIementes,and all thinges of the Elementes compofed, do geue Teftimonie (as Ariftotle fayd) thattheyr Whole Fdijpoftions, yertues , andnaturall motions, depend of the jiffiiuitie of tl heauenly motions and Influences . Whereby, hefide the jftecifcaU order and forme, due to cuery feede: and hefide the Flature ,propre to the Indiuiduall ' Ma • trix, of the thing produced: What fhall he the heauenly Imprefiion,the perfeCf and circumfieffe JFlrologien hath to Conclude. N ot onely (by Apotelefmesft 0 orb but by Nat-urall and Mathematical! demonftratipn Whereunto, what Sciences are reqtiifitc ( without exception ) I partly haue here warned: And In my Propadmmes ( befides other matter there dficlofed ) I haue Mathematically furni- ihed vp the whole Method ; To this our age, not fo carefully handled by any,that euer lohn Dee his Mathematical! Preface. cuer I faw, or heard of. I was, (for * 2i.yeares ago) by certaine earneft difpUtati- * Anno. t 54S ons,of the Learned Gerardus Mercator ,and Antomus Oogaua, (and other, ) therto fo and 1 540. ?;j prouoked:and(by my conftantand inuirtcible zeale to the veritie)in oblcruations Louayn, ofHeauenly Infliienciesf to the Minute of time, )than,fo diligent: And chiefly by the Supernatural! influence,frorn the StarreofIacob,(b directed .-That any Modeft and Sober Student, carefully and diligently fekingforthe Truth, will both finde & cofefie, therin,to be the Veritie, of thefe my wordes: And alfo become a Reafo- nable Reformer, of three Sortes of people: about thefe Influentiall Operations, greatly erring from the truth. Wherof, the one , is Light Beleuers,the other, Note* Light Del|)ilers,and the third Light Pracflifers. The firft,& moft comon Sort, thinke the Heauen and Sterres, to be anfwerable to any their doutes or de- * * fires: which is not fo;and,in dedc,they,to much,ouer reache. The Second forte thinke no Influentiall vertue ( fro the heauenly bodies ) to beare any Sway in Ge- * * neration and Corruption, in this Hlementall world. And to the Sunne , Mone and Sterres ( beingfo many,fo pure,fo bright , fo wonderfull bigge , fo farre in diftancc, fb manifold in their motions , fo conftant in their periodes . &c . ) they affigne a fleight, Ample office or two,and fo allow vnto the(according to their capacities)as much vertue, and power Influentiall,as to the Signe of the Sunne, Mone, and feuen Sterres, hanged vp(for Signesjin Lon don, for diftindion ofhoufes , & fuch groffe helpes,in our wordly affaires: And they vnderftand not(or will not vnderftand) of the other workinges,and vertues of the Heauenly Stinne,Mone, and Sterres .• notfo much, as the Mariner,or Hufband man : no , not fo much, as the Elephant doth, as the Cynocephalus , as the Porpentinc doth : nor will allow thefe perfed , and incor¬ ruptible mighty bodies, fo much v-ertuall Radiation, & Force , as they fee in a litle peece of a Magnes Hone: which, at great diftance,fheweth his operation . And per- chaunce they thinke, the Sea & Riuers ( as the Thames ) to be fome quicke thing, and fo to ebbe,and flow, run in and out, of them felues,at their owne fantafies. God helpe,God helpe. S urely, thefe men,come to fhort : and either are to dull; or willfully blind: or, perhaps, to malicious . The third man, is the common and vulgare '^Aftrologien,ot Pradifer : who, being not duely,artificially,and perfedly 3 . furnifhed:yet,either for vaine glory,or gayne : or like a Ample dolt, & blinde Bay¬ ard, both in matter and maner,erreth:to the difcredit of the Wary , and modeft A- J?roJoricn:cLnd to the robbing of thofe moft noble corporall Creatures,of their Na¬ tural! Vertue: being moft mighty : moft beneficiall to all elementall Generation, Corruptiorfand the appartenances - and moft Harmonious in their Monarchic: For: which thinges, being knowen, and modeftly vfed:we might highly,and conti¬ nually glorifie God, with the princely Prophet, faying. J he Heauens declare the Glorie of GodtSoho made the Heaues in his wife dome: "Soho made the Sonne , for tohaue dominion of the day : the Mone and Sterres to hauc dominion of the nygbt: whereby,® ay today loiter eth talke: and night, to night dedaretb know* ledge.tprayje him, ally e Sterres, and Light. Amen. JN order, nowfoloweth , of Stfttlkc,fomewhat to fay, what we meane by that name: and what commodity, doth, on fuch Art, depend. Statike , is an Arte Mathematicall, which demonflrateth the caufes ofheauynes, and lightnes of all thynges : and of motions and properties , to hea- uynes and lightnes ,belonging. And for afmuch as, by the Bilanx , or Ba¬ lancers the chieffenfible Inftrument , ) Experience of thefe demonftrations may b.iiij. be lohn Dee his Mathematical! Preface. be had: we call this Anfpatike:dn2it isjhe Experimentes of the Balance. Oh, that men wift, what proffit,(aIl maner of wayes)by this Arte might grow, to the hable exa- » miner,and diligent pra&ifer. Thou onely,knoweft all thinges precifely (O God) ■” who haft made weight and Balance, thy Iudgement:. who haft created all thinges in Number ^Waight, and haft way ed themountaines andhilsina Ba- lance: who haft peyfed in thy hand , both Heauen and earth . We therfore war¬ 's ned by the Sacred word, to Confider thy Creatures.-and by that conftderation, to ” Wynne a glyms{asitwere, )or lhaddowof perceiuerance that thy wiledome, >> might, and goodnes is infinite,and vnfpeakable,in thy Creatures declared : And ” being farder adder tiled , by thy mercifull goodnes , that ,three principal! wayes, •• were,ofthe,vfed in Creation ofall thy Creatures , namely , Number^ Waight and '»* OHeafure, And for as much as,of 2N lumber and Meafure, the two Artes(aundent, fa¬ s’ mous,and to humaine vfes moft neceflary, ) are, ail ready, fufficiently kno'wen' and js extant: This third key , we befeche thee ( through thy accuftomed goodnes,) « that it may come to thenedefull and fufficientknowledge,offuch thy Seruauntes, ” as in thy workemanlhip , would gladly finde, thy true occaftons (purpofely of the >» vfed ) whereby we fhould glorifie thy name, and ftiew forth (to the weaklinges in faith) thy wondrous wifedome and Goodnes. Amen. Meruaile nothing at this pang(godly frend,you Geiidc and zelous Student.) An other day, perchaunce, you will perceiue, what occafion moued me. Here, as now, I will giue you fome ground , and withallfomelhew , of certaine commodi¬ ties, by this Arte arifing. And bycaufe this Arte is rare , my Wordes and pradlifes might be to darke : vnleaftyou had fome light, hoiden before the mattenand that, beft will be, in giuing you,out of Archimedes demonftrations,a few principal Con- cluftons,as folowerh. 1. The Superficies ofeuery Liquor, byitfelfe confiflyng ,and in quyet, is Sphxricall : the centre whereof, is the fame , which is the centre of the Earth. 2. . , If Solide Magnitudes, being of the fame bignes, or quatitie, that any Liquor island hauyngalfo the fame Waight : be let downe in¬ to the lame Liquor, they will fettle downeward,fo,that no parte of them,fhall be aboue the Superficies of the Liquor : and yet neuer- theles ,they will not finke vtterly downe,or drowne. V If any Solide Magnitude beyng Lighter then a Liquor , be, let downe into the fame Liquor , it will fettle downe, fo farre into the fame Liquor, that fo great a quantitie of that Liquor, as is the parte of the Solid Magnitude, fettled downe into the fame Liquor : is in Waight, squall, to the waight of the whole Solid Magnitude. 4* Any Solide Magnitude , Lighter then a Liquor , forced downe into lohn Dee his Mathematical! Preface. ifjto the fame Liquor , will moue vpward , with fo great a poweiq by how much , the Liquor hauyng xquall quantitie to the whole Magnitude's heauyer then the fame Magnitude, J- / Any Solid Magnitude, heauyer then a Liquor, beyng let do wne into the fame Liquor, will linke downe vtterly : And wilbe in that Liquor , Lighter by fo much , as is the waight or heauynes of the Liquor, hauing bvgnes or quantitie, squall to the Solid Magnitude, 6, i.V. If any Solide Magnitude , Lighter then a Liquor , be let downe Sphar e according to 1 J ^ j i r r* ^ | | -111 any proportion af- into theiame Liquor , the waight or the lame Magnitudewill be, to the Waight of the Liquor . (Which is aquallin quantitie to the whole Magnitude,) in that proportion , that the parte , of the Mag- nitude fettled downe, is to the whole Magnitude. gY thefe verities , great Errors may be reformed, in Opinion of the Natural! Motion of thinges, Light and Heauy. Which errors, are in Natural! Philofophie (almoft ) of all me allowed.-to much trufting to Authority /and falfe Suppofitions. As, Of any two body es, the heauyer, to moue downward falter then the lighter. This error,is notfirftby me, Noted; but by one lohn Baptist de nedichs. Thechief of his propofitions,is this: which feemeth a Paradox. If there be two bodyes of one forme, and of one kynde, asquail in . quantitie or vnasquall , they will moue by aequall fp ace, in xquall A*ar °X* tymeiSo that both theyrmouynges be in ay re , or both in water ; or in any one Middle. Hereupon , in the feate of Gunny ng,certar e good difeourfes ( otherwife) jV. T. may receiiie great amen dement, and furderance. In the emended purpofe , alio, rj-i . < allowing fomwhat to the imperfection df Nature : not aunfwerable to the preci- fenes oldemon;f ration . Moreouer,by the forefaid propofitiofts ( wifely vied.) thefe Prof oft - The Ayre,the water, the Earth, the Fire, may be nerely,knowen,how light or hea- t ions. uy th y arc { N aturally ) in their affigned partes : or in the whole. And then,to thinges Eiementall,turningvourprad:ife: you may deale for the proportion of the Eiementes , in the thinges Compounded . Then, to the proportions of the Hu¬ mours in Man: their waightes: and the waight of his bones, and fiefh.&c. Than, by waight, to haue confideration of the Force of man, any maner ofwav: in whole or in pai t.Then ,may you, ofShips water drawing , diucrfly,in the Sea and in frefh water, haue plealant confideration and ofwaying vp of any thing, fonken in Sea or mftefli water &c . And (to lift vp your head a loft : ) by waight, you may, as precifely,as by any inftrument els,meafure the Diameters of Sonne and eMone.&c. Frendejpray you , way thefe thinges, with the iuft Balance ofReafon. And you will hnde Memailes vpon Meruailes ; And gfteme one Drop of Truth ( yea in Naturall Phtlo,opnie)more worth, then whole Libraries of Opinions, vndemon- ftrated: or not aunfwenng to Natures Law,and your experience. Leaning thefe c-i* thinges. the waight of the S phare thereto £wymming> A common error jioted* T he practije Statical l, to kriovp the pro¬ portion, be- txvene the Cube, and the Sphare J.D, * For, Jo, bane you. 2 5 6, partes of a Gratae. *The proportion of the Square to the Circle infiribed. *The Squaring, of the Circle,Mecha~ nically . *To any Square geuen, to gene a Circle, equaU. lohn Dee his Mathematical! Preface . thinges,thus:I will giue you two or three,light pra&ifes, to great purpofe : and fb iinifli my Annotation StaticalL In Mathematical! matters , by the Mechanidens ayde , we will behold, here, the Commodity of waight . Make a Cube,of any one Vniforme : and through like heauy ftufFe: of the lame Stuffe,make a Sphere or Globe,precifdy,of a Diameter squall to the Radicall hdc of the Cube . Your ftuffe,may be wood, Copper, Tinne, Lead,SiIuer.&c. (being, as I fayd,oflike na¬ ture , condition, and like waight throughout. ) And you may , by Say Balance, haue prepared a great number of the Imalleft waightes : which, by thofe Balance can be difeerned or tryed.-and fo,haue proceded to make you a perfed Pyle, com¬ pany & Number of waightes: to the waight oflix,eight,or twclue pound waight: moft diligently tryed,all.And of euery one , the Contentknowen, in your lead waight,that is wayable. [They that can not haue thele waightes of precifenes : may, by Sand, Vniforme, and well dufted,makc them a number of waightes, fome- whatnerepredfenes : by hailing euer the Sand : they lliall ,at length, come to a •lead common waight.Therein,I leaue the farder matter, to their difcretion,wiiom nedelliali pinche.] Th e. Venetians conftderation of waight , may feme prccife enough:by eight delcentes progrefsionall,* hailing , from a grayne. Your Cube, Sphere, apt Balance, and conuenient waightes,being ready -fall to worker . Firft, way your Cube.Note the Number of the waight . Wav, after that, your Sphere. N ore likewife,th e N uber of the waight.If yo u now find the waight ofyour C ube, to be to the waight of the Sph^re,as 21 . is to n.-Then you fee, how the Mechani- cien and Experimenter , without Geometrie and Demonftration, are ( as nerely in effed)£ought the proportion of the Cube to the Sphere : as I haue demonftrated it,in the end of the twelfth boke of Euclide. Often , try with the lame Cube and SphcCre.Then,chaunge,your Sphere and Cube,to an other matter: or to an other bignes : till you haue made a perfed vniuerfall Experience ofit. Pofsible it is, that you lliall wynne to nerer termes,in the proportion. When you haue found this one certaine Drop of Naturall veritic,procede on, to Inferre,and duely to make allay, of matter depending. As, bycaule it is well de¬ monftrated , that a Cylinder , whofe heith , and Diameter of his bafe,is squall to the Diameter of the Sphere ,is Sefquialterto the fame Sphere (thatis,as 3. to 2:) To the n umber of the waight of the Sphere, adde halfe fo much, as it is : and fo haue you the number of the waight of that Cylinder. Which is alfo Compre¬ hended of our former Cube: So,that the bafe of that Cylinder , is a Circle deferi- bed in the Square , wrhich is the bafe of our Cube. But the Cube and the Cy- linder, being both of one heith , haue their Bafes in the fame proportion , in the which, they are , one to an other, in their Mafsines or Soliditie . But,before,we haue two numbers, exprelsing their Mafsines , Solidities , and Quantities , by Waight :wherfo re, we haue * the proportion of the Square,to the Circle, inlcribed in the fame Square. And fo are we fallen into the knowledge fenlible, and Expe¬ rimental! of ^Archimedes great Secret: of him , by great trauaile of minde , fought and found. WherFore,to any Circle giuen , you can giue a Square squall : * as I haue taught, in my Annotation, vpon the lirft propolition of the twelfth boke. And likewile,to any Square giuen, you may giue a Circle squall: *Ifyou deferibe a Circle,which lliall be in that proportion, to your Circle inlcribed, as the Square is to the fame Circle -This,you may do,by my Annotations, vpon the lecond pro¬ polition of the twelfth boke of Euclide , in my third Probleme there. Your dili¬ gence may come to a proportion, of the Square to the Circle inlcribed , nerer the tmth,then is the proportion of 14.10 n. And confider, that you may begyn at the Circle and Square, and fo come to conclude of the Sph^re,& the Cube, what John Dee his MathematicathBr.xface. their proportion is:as now , yon came from the Sphere to the Circle. For, of Sib uer,or Gold, or Latton Lamyns or plates (thorough one hole drawees the manef is)if you make a Square figure.-& way itrand then,defcrib.ing theron, the Circle in fcribed:& cut of,& file away,precifely (to the Circle) the ouerplus of the Square: you lhall then,waying your Circle , fee, whether the waight of the Square , be to your Circle, as 14. ton. As I haue Noted , in the beginning of Cuchdes twelfth boke.&c.after this refort to my laft propofition,vpon the laft of the twelfth . An d there, helpe your felfe, to the end. And, here, Note this, by the -way . That we may Square the Circle , without hauing knowledge of the proportion, of the Cir¬ cumference to the Diameter : as you haue here perceiued . And otherwayes allb, I can demonftrate it.So that, many haue cumberd them felues fuperfluoiifly, by trauailing in that point firft , which was not of neeefsitie,fijrft : and alfb very in¬ tricate. And eafily,you may, (and that diuerfly) come to the knowledge of the Circumference: the Circles Quantitie , being firft knowen. Which thing,! leaue to your confideratiommaking haft to defpatch an other Magiftrall Probleme: and to bring it,nerer to your knowledge,and readier dealing with, then the world (be¬ fore this day,) had it for you, that I can tell of. And that is, A Mechmicall Dubblyng of the Cube:&c. Which may, thus, be done: Make of Copper plates, orTyn plates, a fourfquare vpright Pyramis,or a Cone: perfectly fafhioned in the holow, within . Wlierin, let great diligence be yfed , to ap~ proche (as nere as may be ) to tbe Mathematical! perfection of thofe figures . At their bafes, let them be all open : euery where, els, moll dole, and iufl to. From the vertex, to the Circumference of the bafe of the Cone: & to the fides of the bafe of the Pyramis :Let q.flraight lines be drawen,in the infide of the Cone and Pyramis : makyng at their fall, on the perimeters of the bafes , equall angles on both fides them felues , with the fayd perimeters . Thefe 4. lines ( in the Pyra¬ mis :andas many, in theCone)diuide:one,in 12. ^quallpartes : and an other, in 24. an other, in 60 , and an other, in 100 . (reckenyng vp from the vertex. ) Or vfe other numbers of diuifion , as experience fh all teach you. Then,* fet your Cone or Pyramis, with the vertex downward , perpendicularly , in refped of the Bafe. (Though it be otherwayes, it hindreth nothyng.) So let the moll fledily be flayed. N ow, if there be a Cube, which youwold haue Dubbled.Make you a prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or Cubik coffen, of Copper, Siluer, Tynne,or Wood &c . Thefe, you may fo proportio in refpeft ofyour Pyramis or Cone , that the Pyramis or Cone, will be liable' to conteine the waight of them, in water, 3 .or 4. times :at the leaft: what ftufffo euer they be made of.Let not your Solid angle , at the vertex,be to lharpe: but that the water may come with eafe,to the very vertex,of your hollow Cone or Pyramis.Put one ofyour Solid Cubes in a Balance apt: take the waight therof ex¬ actly in water . Powre that water, ( without Ioffe ) into the hollow Pyramis or Cone>quietly. Marke in your lines, what numbers the water Cutteth : Take the waight of the lame Cube again e .• in the lame kinde of water , which you had be¬ fore : put that * alfo, into the Pyramis or Cone, where you did put the firft. Marke now againe, in what number or place of the lines, the water Cutteth them. Two c.ij. wayes Note Squaring of the Circle "yitbotuknowt ledos of the proportion ve- tveene Cir¬ cumference and'Diame- ter. ToOubble the Cube re - dilj : by Art - Afechanicall : depending vp- 1 pen ‘Demon- - ilration Add- thematic all. 1. D. T he ^..fidos of this Tyramis muft be 4. Ifofceies Triangles 4 like mdaqmU. T.D. * In all workinget with this Pyramis or Cone, Let their Situations be in all Pointes and Condi¬ tions, a like, or all anetwhileyouare about one worke.Elf you will ene. I.D. * Conjider well whan you muft put your waters togyther: and whan, you muft emp¬ ty your first water, out of your Pyramis or Cone. Els yeti WiU erre. * Vlitruuius. Lib.p.Cap.y, GT God be than¬ ked for this Jnuention,dr thefrnite en- -*Note. Note, .as con¬ cerning the Spharicall Superficies of the Water. KF » >? ?? Ji JJ *Note. Note this A- bridgement of Dabbling the pube.&C' Iohn D ee his Mathematical! Preface. wayes you may conclude your piirpofe : it is to wete , either by numbers or lines. By numbers : as, if you diuide the fide of your Fundamental! Cube into fo many squall partes, as it is capable of,conueniently,with your cafe , and pre- cifenes of the diuifion . For, as the number of your firft and leffe line ( in your hollow Pyramis or Cone,) is to the fecond or greater ( both being counted from the vertex) fofhall the number of the fide of your Fundamentall Cube, be to the nuber belonging to the Radicall fide, of the Cube, dubble to your Fun- dam entail Cube: Which being multiplied Cubik wife, will fone fhew it felfe, whe¬ ther it be dubble or no , to the Cubik number ofyour Fundamentall Cube . By lines, thus; As yoiir lefleand firft line, (in your hollow Pyramis or Cone,)is to the fecond or greater, fo let the Radicalfide ofyour Fundametall Cube, be to a fourth proportionall line , by the 1 2 . propofition, of the fixth boke of Euclide . Which fourth line,fhall be the Rote Cubilqor Radicallfide of the Cube , dubble to your Fundamentall Cube : which is the thing we defired . For this, may I ( with ioy) lay, eyphka, eyphka, eyphka : thanking the holy and glorious Trinity: hauing greater caufe therto , then * t Archimedes had (for finding the fraude vfed in the Kinges Crowne, of Gold) : as all men may eafily ludge : by the diuerfitie of the frute following of the one, and the other . Where I fipakc before, of a hollow Cu¬ bik C offen.-the like vle,is ofit: and without waight.Thus. Fill it with water, preci- fely full,and poure that water into your Pyramis or Cone, And here note the lines cutting in your Pyramis or Cone . Againe,fill your coffen,like as you did before. Put that Water,alfo,to the firft . Marke the fecond cutting of your lines . N ow, as you proceded before, fo muft you here precede . * And if the Cube, which you lb ou Id Double, be ncuer lb great ; you haue, thus, the proportion (in ftnall ) be- twene your two litle C ubes : And then, the fide, of that great Cube(to be doubled) being the third , will haue the fourth, found, to it proportionall : by the 12. of the fixth ofEuclide. N ote,that all this while,”! forget not my firft Propofition Staticall,here rehear- fed: that, the Superficies of the water, is Spharicall . Wherein, vfe your diferetion: to the firft line,addinga finall heare breadth,more:and to the fecond,halfe a heare breadth more, to his length . For, you will eafily perceaue, that the difference can be no greater, in any Pyramis or Cone, of you to be handled. Which you fhall thus try e . F or finding the fwelling of the water aboue leuell . Square the Semidiame¬ ter, from the Centre of the earth,to your firft Waters Superficies . Square then, halfe the Subtendent ofthat watry Superficies ( which Subtendentmuft haue the equall partes of his meafure, all one, with thofe of the Semidiameter of the earth to your watry Superficies) : Sub trade this fquare,ffom the firft: Of the refidue, take the Rote Square. That Rote, Subtrade from your firft Semidiameter of the earth to your watry Superficies : that, which remaineth, is the heith of the water, in the middle, aboue the leuell . Which, you will finde, to be a thing infenfible. And though it were greatly fenfible,* yet, by helpe of my fixtTheoreme vpon the laft Propofition of Euclides twelfth booke, noted : you may reduce all, to a true Leuell . But, farther diligence,of you is to be vfed,againft accidentall caufes of the waters fwelling:as by hauing(fomwhat)with amoyft Sponge,before,made moyft your hollow Pyramis or Cone, will preuent an accidentall caufe of Swelling, &c. Experience will teach you abundantly : with great eafe, pleafure,and comoditie. Thus,may you Double the Cube Mechanically, Treble it, and fo forth, in any proportion . N ow will I Abridge your paine, coft, and Care herein. Without all preparing ofyour Fundamentall Cubes : you may (alike) worke this Conclufion. For, that, was rather akinde of Experimentall dem6ftration,then the fhorteft way: and lohn Dee his Mathematical! Er^face. and all, vpon one Mathematicall Demonftration depending . Take water ( as much as conueniently will ferue your turne : as I warned before of your Funda¬ mental! Cubes bignes ) Way it precifely . Put that water, into your Pyramis or Cone . Of the lame kinde of water, then take againe, the lame waight you had before : put that fikewile into the Pyramis or Cone . For, in eche time, your mar¬ king of the lines, how the Water doth cut them, Ibail geue you the proportion be- twen the Radicall fides,crf any two Cubes, wherof the one is Double to the other: working as before I haue taught you:*lauing that for you Fundamental! Cube his Radicall fide: here,you may take a right line, at pleafure. Yet farther preceding with ourdroppeof Naturall truth : you may (now) geue Cubes, one to the other, in any proportio geue: Rational! orlr- rationall : on this maner.Make a hollow Parallelipipedon of Copper or Tinn'c: with one Bafe wating, or open:as in our Cubike Coffen. Fro the bottomc of that Paralielipipedon,raife vp,many perpendiculars, in euery ofhis fower fides.Now if any proportion be alfigned you, in right lines. -Cut one ofyour perpendiculars (or aline equall to it , or lefie then it ) likewife : by the lo.of the fixth of Eu elide. And thofe two partes , fet in two fundry lines of thofe perpendiculars ( or you may fet them both, in one line) making their bcginninges,to be, at the bale: and fo their lengthes to extend vpward . Now, fet your hollow Parallelipipedon, vpright, perpendicularly,fteadie . Poure in water, handfomly,totheheith ofyour Ihorter line . Poure that water, into the hollow Pyramis or Cone . Marke the place of the riling. Settle your hollow Parallelipipedon againe . Poure water into it: vnto the heithofthe fecond line , exa&ly . Poure that water * duely into the hollow Pyramis or Cone : Marke now againe, where the water cutteth the lame line whichyoumarkedbefore . For, there, as the firft marked line, is to thefe- cond ; So fhall the two. Radicall fides be, one to the other, of any two Cubes: which, in their Soliditie, lhall haue the lame proportion, which, was at the firft afi figned : wereitRationali or Irrationall .• Thus, in fundry waies you may furnilhe your felfe with fuch ftraungc and pro¬ fitable matter: which, long hath bene wifhed for. And though it be Naturally done and Mechanically : yet hath it a good Demonftration Mathematicall . Which is this .• Alwaies ,you haue two Like Pyramids : or two Like Cones,in the proporti¬ ons alfigned : and like Pyramids or Cones,are in proportion,one to the other, in theproportion of their Homologall fides (or lines) tripled. Wherefore, if to the firft, and fecond lines,foun din your hollow Pyramis or Cone, you ioyne a third and a fourth, in continuallpropbrtion .- that fourth line, lhall be to the firft, as the greater Pyramis or Cone, is to the lelfe : by the 33-of the eleuenth ofEuclide . If Pyramis to Pyramis, or Cone to Cone, be double , then lhall * Line to Line, be alfo double, &c. But,as our firft line, is to the fecond, fo is the Radicall fide of our Fundamentall Cube,to the Radicall fide of the Cube to be made , or to be dou¬ bled: and therefore, to thofe twaine alfo, a third and a fourth line , in continuall proportion, ioyned : will geue the fourth line in that proportion to the firft,as our fourth Pyramidall, or Conike line, was to his firft : but that was double, or tre- ble,&c.as the Pyramids or Cones were, one to an other (as we haue proued) thcr- fore,this fourth, ihalbe alfo double or treble to the firft,as the Pyramids or Cones were one to an other .* But our made Cube,is deferibed of the fecond in proporti¬ on, of the fower proportionall lines : therfore * as the fourth line, is to the firft, lb is that Cube, to the firft Cube : and we haue proued the fourth line, to be to the firft, as the Pyramis or Cone, is to the Pyramis or Cone Wherefore the Cube is c.iij. to the n yy yy yy yy }} yy T ogme Cubes one to the o'— t her in any proportion. Ratio nail or Irrationall. yy yy >t it j? }> ?> ?> » yy » ' Empty¬ ing the firft. The demonfiratient vf this Dubbling of the Cube And of the reft. r.D. * Hereby ,helpe you, filfto become a pr, cife praSHftr. ^Ant, fo con ftder.horo, no¬ thing at all . you ar hindred(ftnftbly ) , the Conuexitie of the water. rBy the 3!.cf the e leuenth bookc of Euclide . l.T>. *Undy out diligence in praftije.can Jo f in -tonight ofiva- terjpeiforme it: Therefore, now. you are able to geue good reafon of your whole doing. * Note this Corollary. *T he great Commodities following of thefe new In¬ ventions. Such is the Fruiteofthe Mathemati - tali Sciences and Artes ♦ lohn Dee his Mathematicall Preface. to the Cube, as Pyramis is to Pyramis, or Cone is to Cone . But we * Suppofc Py- ramis to Pyramis,or Cone to Cone, to be double or treble.&c. Therfore Cube, is to Cube,double,or treble, &c.Which was to be demonftrated. And of the Paralle- lipipedo,it is euidet , that the water Solide Parallelipipedons,are one to the other, as their heithes are,feing they haue one bale . Wherfore the Pyramids or Cones, made of thofe water Parallelipipedons,are one to the other, as the lines arefone to the other) betwene which,our proportion was affigned . But the Cubes made of lines, after the proportio of the Pyramidal or Conik homologall lines, are one to the other,as the Pyramides or Cones are , one to the other ( as we before did proue) therfore, the Cubes made, flralbe one to the other,as the lines alfigned,are one to the othen'Which was to be demonftrated.Note. *This,my Demonftratiois more generall,then onely in Square Pyramis or Cone: Confider well . Thus , haue I, both Mathematically and Mechanically, ben very long in wordesiyet fl truft)no- thing tedious to thcm,who, to thefe thinges , are well affedied. And verily I am forced (auoiding prolixitiejto omit fundry fuch things, ealie to be pradifed: which to the Mathematicien,would be a great Threafure : and to the Mechanicien,no fmall gaine.*N ow may you, Betwene two lines ginen ,finde two middle proportionals, in Continuall proportion : by the hollow Paralleli- pipedon, and the hollow Pyramis,. or .Cone.. Now,anyParallelipipedon rectangle being giuen:thre right lines may be found,proportionall in any propor¬ tion aiTigned,ofwhich,ilial be produced a Parallelipipedon, a?quall to the Paralle- lipipedon giuen.Hereof,I noted fomwhat,vpon the 36,propolition,ofthe n.boke of Euclide. N ow,all thofe thinges, which Vitruuius in his Archite<fture,fpecified liable to be done, by dubbling of the C ube Or, by finding of two-middle propor- tionall lines, betwene two lines giuen,may eafely be performed . Now,thatPro- bleme, which I noted vnto you, in the end ofmy Addition, vpon the 34.ofthe n. boke of Euclide , is proued pofsible. N ow,may . any regular body, be T ranflormed into an other, &c. N ow, any regular body : any Sphere, yea any Mixt Solid : and (that more is)Irregular Solides, may be made(in any proportio affigned)like vnto the body, firft giuen. Thus, of a Manneken, (as the Dutch Painters terme it)in the fame Symmetric , may a Giant be made.- and that, with any gefture,by the Manner ken vfed : and contrary wife.N o w, may you , of any Mould, or Model! of a Ship, make one,of the lame Mould (in any affigned proportion) bigger or lelfer. Now, may you,of any*Gunne,or little peece or ordinauce,make an other,with the fame Symmetric ( in all pointes) as great, and as little,as you will.Marke thatr and thinke on it , Infinitely , may you apply this, fo long fought for, and now fo eafily concluded : and withall,fo willingly and frankly communi¬ cated to fuch, as faithfully deale with vertuous ftudies.Thus,canthe Mathematicall minde, deale Speculatiuely in his own Arte: and by good meanes. Mount aboue the cloudes and fterres : And thirdly, he can, by order, Defcend,to frame Naturall thinges, to wonderful! vfes :and when he lift , retire home into his owne Centre : and there,prepare more Meanes, to Afcend or Defcend by : and, all,to the glory of God , and our honeft delegation in earth. Although, the Printer , hath lookedfor this Preface, a day or two , yet could I not bring my pen from the paper , before I had giuen you comfortable warning, and brief inftru<ftion s,offome of the Commodities,by Statike^ hable to be reaped: In the reft,I will therfore,be as brief, as it is pofsible/and with all,defcribing them, fomwhat accordingly. And that,you fhall percciue,by this, which in order com- v ineth lohn Dee his Mathematical! Preface. meth next. For,wheras, it is fo ample and wonderfu 11, that, an whole yeare long, one might finde fruitfull matter therin,to fpeake of:and alfo in pra&ile3is a Threa- fure endeles :yet will I glanfe ouer it, with wordes very few. This do I call Anthropographte. which is an Art reftored , and of my preferment to your Seruice. I pray you, thinke of it , as of one of the chief pointes,of Humane knowledge. Although itbe,butnow,firft Cofirmed, with this new name : yet the matter, hath from the beginning, ben in confederation of all perfed P hilofophers. Anthropographie,is the defcription of the Num¬ ber ,Meafure, Waight , figure, Situation, and colour of euery diuerfe thing, conteyned in the perfect body of MAN : with certain know¬ ledge of the Symmetric , figure , waight , Characterization, and due locall motion, of any parcell of the fayd body, afsigned; and of N fi¬ bers, to the fayd parcell appertainyng. This,is the onepartofthe Defini¬ tion, mete for this place: Sufficient to notifie, the particularitie, and excellency of the Arte:and why itis, here , afcribed to the Mathematical. Yf the defcription of the heauenly part of the world,had a peculier Art,called o Ajlronomie If the de¬ fcription of the earthly Globe, hath his peculier arte,call ed Geographic. If the Mat¬ ching ofboth,hath his peculier Arte, called Cofmographic : Which is the Defcriptid of the whole,and vniuerfall frame of the world : Why fliould not the defcription of him,who is the Leife world:and,fro the beginning,called CVticrocofmtts ( that is. M JJT he Lefte World.)kn& for whole lake, and feruice,all bodily creatures els, were t^oe LcIfe created : Who,alfo,participateth with Spirites, and Angels : and is made to the I- W&rld‘ mage and fimilitude of GW: haue his peculier Artc'and be called the frteofArtesi rather, then, either to want a name,or to haue to bale and impropre a name i You muft offundry profeffions,borow or challenge home , peculierpartes hereofrand farder procede: as, God, Nature , Reafon and Experience lhall informe you. The Anatomiftes willreftore to you,fome part: The Phyfiognomiftes,fome:The Chy- romantiftes fome.The Metapofcopiftes,fome: The excellent, Albert Durer,*. good part: the Arte ofPerfpediue,will fomwhat,for the Eye,helpe forward : Pythagoras , Hipocrates, Plato,Galenus,Meletius,& many other (in certaine thinges ) will be Con- tributaries. And farder, the Heauen,the Earth, and all other Creatures, will eche fiew, and offer their Harmonious feruice , to fill vp, that, which wanteth hereof: and with your own Experience, concluding : you may Methodically regifter the whole, for the pofteritie : Whereby, good profc will be had, of our Harmonious, and Microcofinicall conftitution. The outward Image,and vew hereof: to the Art of Zographie and Painting, to Sculpture , and 'Archite&ure (for Church,Houfe, Mwocofi Fort, or Ship) is moft necefaiy and profitable : for that, itis the chiefe bafe and mm. foundation ofthem . Lookein * Vitruuius, whether I deale fincerely foryour * Lib y* behoufe, or no . Lookein Alhertus D ur er us, De Symmetriahumani Corporis. Looke ^ap • 1 • in the 2 7. and 28. Chapters, of the lecond booke, De occulta Philofophia . Confi- der th eArke of Hoe . And by that, wade farther . Remember the Delphicall Oracle 2i0 S C E T El P S v M ( Kpnoive thy felfe ) fo long agoe pronounced : of fo many a Philofopher repeated : and of the Wifett attempted : And then, you will perceaue, how long agoe, you haue bene called to the Schole,where this Arte might be learned. Well. Iam nothing affrayde,ofthedildayneoflbme Rich, as thinke Sciences and Artes, to be but Seuen. Perhaps, thole Such, may, with igno¬ rance, and lhame enough, come fliort ofthem Seuen alio : and yet neuerthelefe c.iiij. they lohn Dee his Mathematical! Preface, they can not prcfcribe a certaine number of Artesrand in eche, certaine vnpaflable boundes,ro God,Naturc,and mans Induftrie.New Artes,dayly rife vp: and there O* was no fuch order taken, that, All Artes,lbould in one age, or in one land, or of one man,be made knowen tothe world.Let vs embrace the giftes of God , and wayes to wifedome , in this time of grace , from aboue , continually bellowed on them., who thankef ully will receiue them : Et bonis Omnia Ccoperabuntur in bonum. Trochilike, is that Art Mathematical!, which demonftrateth. theproperties of all Circular motions , Simple and Compounde. And bycaufe the frute hereofvulgarly receiued,is in Wheles , it hath the name of Trochilike : as a man would lay ,Whele Art. By this art, a Whele may be geuen which lhall moue ones about, in any tyme aligned . Two Wheles may be giuen, whofe turnynges about in one and the fame tyme, ( or equall tymes) , lhall haue, one to the other, any proportionappointed . By Wheles, may a. ftraight line be defcribed : Likewife,a Spirail line in plaine,Conicali Se&ion Iines,and other Irre¬ gular lines, at pleafure, may be drawen . Thefe, and fuch like, are principal! Con- clulions of this Arte : and helpe forward many plealant and profitable Mechani- Saw Milks, call workcs : As Milles,to Saw great and very long Deale bordes , no man being by . Suchhaue I feenein Germany : and in the Citieof Prage ; in thekingdome of Bohemia : Coyning Milles,Hand Millesfor Corne grinding: Andallmaner of Milles,and Whele worke: By Winde, Smoke, Water, Waight, Spring, Man or Beall, moued . Take in your hcmd,Agricola here CAEctalUca : and then lhall you (in all Mines) perceaue,how great nede is, ofWhele worke.By Wheles,llraunge workes and in credible, are done - as will, in other Artes hereafter, appeare. A won. derfull example of farther polfibilitie, and prefent commoditie , was fene in my time, in a certaine Inllrument: which by the Inuenter and Artificer(before) was folde for xx. Talentes of Golde:and then had (by milFortune)receaued fome iniu- rie and hurt : And one lanellm of Cremona did mend the lame, and prefented it vn- to the Emperour Charles the fifth . Hieronymus Cardanus , can be my witnelle, that therein, was one Whele, which moued, and that,in fuch rate,thar,in yoco.yeares onely, his owne periode fliould be finifiied . A thing almoffc incredible : But how farre,l keepe me within my boundcs: very many men(yetaliue) can tell. HellCofophlC, is nere Siller to Trochilike: and is. An Arte Mathema¬ tically which demonftrateth the defigning of all Spirail lines in Plaine , on Cylinder , Cone , Sphere , Conoid , and Spheroid, and their properties appertayning . The vfe hereof, in Architecture , and di- uerfe Inflrumentes and Engines, is moll necellary. For,in many thinges,the Skrue worketh the feate, which, els, could not be performed . By helpe hereof , it is *Atheneus * recorded, that, where all the power of the Citie of Syracufa,was not hableto Ltb. 5 .cap, 2. moue a certaine Shipfbeing on ground)mightie Archimedes , letting to , his Skruifh Engine , caufed Hiero the king , by him felf , at eafe,to remoue her, as he would. Trodu. Wherat,the King wondring : A?ro mans acm-os, ApyjAi Aycm-msAo/s. Pag. r 8 . From this day, forward (laid the King ) Credit ought to be giuen to Archimedes , what focuer he fayth. Pneumatithmie demonflrateth by clofe hollow Geometri¬ cal! Figures,(regular and irregular ) theftraunge properties ( in mo¬ tion or ftay)of the Water, Ayre, Smoke , and Fire,in theyr cotinuitie, and Iohn Dee his Mathematical! Preface. and as they are ioyned to the Elementes next them. This Arte , to the Naturall Philofopher,is very profitable: to proue,that Vacuum , or Emetines is not in the world. And that, all N attire, abhorreth it fo much : that , contrary to ordi¬ nary law,the Elementes will moue or hand . As, Water to afcend: rather then be- twene him and Ayre,Space or place lhould be left,mOre then (naturally) that qua- titie of Ayre requireth,or can fill. Againe, Water to hang, and not defcend: rather then by defcending,to leaue Eiriptines at his backe . The like, is of Fire and Ayre: they willdefcend.-when,cither, their Cotinuitiefhould be dif!blued:or their next Element forced from them. And as they willnotbeextended,todifcontinuitie: So, will they not, nor yet of mans force,canbe preft or pent,in fpace , notfufhcient and.aunfwcrable to their bodily fubftance.Great force and violence will they vfe, to enioy their naturall right and Iibertie. Hereupon, two or three men together, bykeping Ayrevnder a great Cauldron, and forcyng the fame downe, orderly, may without harme defcend to the Sea bottome : and continue there a tyme &c. Where,N ote,how the thicker Element(as the Water)giueth place to the thynner (as, is the ayre:)and receiueth violence of the thinner, in maner. Sec. Pumps and all maner of Bello wes, haue their ground of this Art: and many other ftralinge dc- uifes. Asflydraulica&rganes goyng by water. Sec. Of this Feat, (called common¬ ly Pneumatica> ) goodly workes are extant , both in Greke,and Latin . With old and learned S chole m en,it is called S dentin de pleno & vacuo. Menadrie, isan Arte Mathematicall, which demonfirateth, how , aboue Natures vertue and power fimple : Vertue and force may be multiplied : and fo, to dired,to lift, to pull to , and to put or caft fro , any multiplied or fimple , determined Vertue , Waight or Force: naturally, not, fo , diredible or moueable. Very much is this Art furdred by other Artes • as, in fome poihtes, by Perjfeciiue: in fome, by St at ike : in fome,by Trochili ke.znd in othcr,by Helicofophie.-znd Pneumatithmie. By this Art, all Cranes, Gybbettes,& Ingines to lift vp , or to force any thing, any maner way, are ordred : and the certaine caufe of their force, is knoWne : As, the force which one man hath with the Duche waghen Racke:therwith,to fet vp agayne,a mighty waghenladen,befng ouerthrowne. The force of the Croflebow Racke, is certain¬ ly, here,demonftrated.The reafon,why one m5, doth with a Ieauer,Iift that8which Sixe men, with their handes onely, could not, fo eafily do. By this Arte,in ottr common Cranes in London , wherepowreisto Cranevp,thewaight df 2000. pound: by two Wheles more (by good order added ) Arte concluded!, that there may be Craned vp 20oooo.pound waight &c.So well knew Archimedes this Arte.* that he alone, with his deuifes and engynes,(twife or thrife)fpoyIed and difeomfi- ted the whole Army and Hofte of the Romaines, befieging Syracufa, Marcus Mar- cellus the Confed, being their Generali Capitaine. Such huge Stones, fo many, with fuch force , and fo farre , did he with his engynes hay le among them, out of the Citie. And by Sea likewife : though their Ships might come to the walls ot'Syra- cufa , yet hee vtterly confounded the Romaine Nauye. > What with his mighty Stones hurlyng: what with Pikes of* / 8 fote long, made like lhaftes: which he for¬ ced almoft a quarter of a myle.What, with his catchy ng hold of their Shyps , and hoyfing them vp aboue the water , and fuddenly letting them fail into the Sea a- gaine:what with his* Burning Glades; by which he fired their other Shippcs afar- of: what, with his other pollicies, deuifes, and engines, he fo manfully acquit him felfe : that all the Fotce,courage,and pollicie of the Romaines (for a great feafon) d.j. could To goto the bottom of the Sea Without damoer. o Plutarchus in Mir* eo Marcello . Synejiusia Epifio * lis. Polybius* Plinius. QuintilUnus. T. Liuius. *iAthen&uf, * Qalenus. o xAnthemtus. ‘Burning GLtJfes. Games* lohn Dee his Matheifiatkall Preface . could nothing prcuailc/or the winning ofSyracula. Wherupon , the Romanes named Archimedes, Briareus^nd Cmtimmus. Zonaras maketh mention of one Pro-* clftSj who fo well had perceiued Archimedes Arte of c Memdrie , and had fb well in* uented of his owne , that with his Burning Glades, being placed vpon the walks of Byfance , he multiplied lb the heate of the Sunne,and direded the bcames of the lame again ft his enemies Nauie with fuch force , and fo fodeinly ( like lighte- ning)that he burned and deftroyed both man and ihip . And Dion fpecifieth of Prijcits, a Geometrickn in Bylance, who inuented and vied fondry Engins, of Force multiplied : Which was caule, that the Empcroxr Seuerus pardoned him, his lifc,af* ter he had wonne BylancecBycaufe he honored the Arte , wytt, and rare induftrie of P rife ft s. But nothing inferior to the inuention of thefe engines of Force, was the inuention ofGunnes. Which, from an Englifh man, had the occalion and order of firft inuenting: though in an other Iand,and by other men,it was firft cxecuted.- And they thatlhould fee the record, where the occalion and order general!, of 3) Gunning, is firft difcourfed of,wouid thinke: that,linall thinges,llight5and comon : 3, commingto wife mens confideration,and induftrious mens handling , may grow ,3 to be of force incredible. Hypogciodie, is an Arte Mathematical!, demonftratyng,how, vnder the Sphxricall Superficies of the earth, at any depth, to any perpendicular line afsigned(whofe diftance from the perpendicular of the entrance: and the Azimuth ,likewife, in refped: of the faid en¬ trance, is knowen) certaine way may be prarfcribed and gone : And how, any way aboue the Superficies of the earth deligned , may vn~ d^r earth, at any depth limited , be kept : goyng alwayes , perpendi¬ cularly ,vnder the way , on earth defigned : And, contrarywife,Any way,(ftraight or croked , )vnder the earth, beynggiuen : vppon the vtface, or Superficies of the earth, to Lyne out the fame : So, as, from the Centre of the earth , perpendiculars drawen to the Spherical! Supefficies of the earth, fhail precifely fall in the Correfpondent poirites of thofe two wayes . This , with all other Cafes and cir- cumftances herein , and appertenances , this Arte demonftrateth . This Arte, is very ample in vanetie of Conclusions : and very profitable fundry wayes to the Common Wealth . The occafion of my Inuenting this Arte, was at the requeft of two Gentlemen, who had a certaine worke( of gaine) vnder ground: and their groundes did ioyne ouer the worke : and by reafon of the crokednes, diuers depthes, and heithes of the way vnder ground , they were in doubt, and at controuerfie, vnder whofe ground, as then, the worke was * The name onely (be¬ fore this ) was of me publifhed, Deltinere Snbtcrraneo The reft, be at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole, and for fecretepaffages vnder ground, betwene place and place (as this land hath diuerie) and for other purpofes^any man may eafily perceaue, both the great fruite of this Arte, and alfo in this Arte, the great aide of Geometric. Hydragogie, demonftrateth the poffible leading of Water, by Natures lawe , and by artificial helpe , from any head (being a Spring, {landing, or running Water ) to any other place affigned. Long Iohn Dee his Mathematical! Preface, Long, hath this Arte bene in vfe : and much thereof written : and very marueilous workes therein, performed : as may yetappeare,in Italy :by the Kuynes remaining of the Aquedu&es . In other places, of Riuers leading through the Maine land* Nauigable many a Mile. And in other places,of the marueilous forcinges of Wa¬ ter to Afcend . which all, declare the great fkiIl,to be required ofhim,who fhould in this Arte be perfe&c, for all occafions of waters poftlble leading . T o Ipeakc of the allowance of the Fall, for euery hundred foote: or of the Ventills (if the wa¬ ters labour be farre,and great) I neede not .• Seing, at hand (about vs)many expert men can fufficiently teftific, in effe&c, the order : though the Demonftration of the N eceifitie thereof* they know not : N or yet , if they lhould be led* vp and downe, and about Mountaines, from the head of the Spring: and then, a place be¬ ing afligned : and of them, to be demaunded, how low or high, that laft place is, in refpe&e of the head, from which (fo crokediy, and vp and downe ) they be comes Perhaps,they would not, or could not, very redily,or nerely afloyle that quefrion. Geometric therefore, is neceflary to Hydragogie . Of the fimdry wayes to force wa¬ ter to afcend , eyther by T ympane, Kettell mills, Skrue , Cteftbike , or filch like : in Vi~ truuim, ^Agncola, (and other,) fully, the maner may appeare . And fo, thereby ,alfb be m oft euident, how the Artes, of P neumatithmie, Heiicofophie, Statike , Trochihke , and CMenadrie , come to the furniture of this,in Speculation, and to the Corn mo¬ di tie of the Common Wealth,in praftife. Horometrie, is an Arte Mathematical!, which demofbrateth, how, at all times appointed, theprecife vfuall denominatio of time, may be knowen, for any place afligned . Thefe wordes,are finoth and plaine ealie Englilhe, but the reach of their meaning, is farther, then you woulde lightly imagine . Some part of this Arte, was called in olde time, Gnomomce: and of lat eflorologiographia : and in Englilhe, may be term ^Dulling . Auncient is the vfe , and more auncient,is the Inuention . The vfe,doth well appeare to haue bene (at the Ieaft) aboue two thoufand and three hundred yeare agoe in * King ^Reg.zo* K^dchaz, Diall, then,by the Sunne,lhewing the diftin&ion of time . By Sunne, Mone,and Sterres,this Dialling may be performed, and the precife Time of day or night kno wen . But the demonftratiue delineation of thefe Dialls,of all fortes* requireth good fkill, both of AHronomie, and Elementall, SphccricalfPhx- nomenaII,and Conikall . Then, to vfe the groundes of the Arte, for any regular Superficies, in any place offred : and ( in any poflible aptpofition therof) theron, to defcribe ( all maner of wayes ) how, vfualLho wers, may be ( by the $ times flia- dow ) truely determined : will be found no Height Painters worke . So to Paint* and prefcribe the Sunnes Motion, to the breadth ofaheare. In this Feate(in my youth ) I Inuented a way, How in any Horizontall,Murall,or ./Equine- dtiall Diall, &c. At all howers (the Sunne fhining) the Signe and De- gree afcendent,may be knowen . which is a thing vetf neceffary for the Riling of thole fixed Sterres : whofe Operation in the Ayre, is of great might, euidently . I Ipeake no further, of the vfe hereof. But forafmuch as, Mans affaires require knowledge of Times & Momentes,when,neither Sunne,Mone,or Sterre, can be fene: Therefore, by Induftrie Mechanical!, was inuented,firft,how,by Wa- ter,running orderly, the Time and howers mightbe knowen: whereof, theVamous Cteftbms , was Inuentor : a man, of Vitruuius, to the Skie (iuftly) extolled . Then, after that, by Sand running,were howers' meafured : Then,bv TrocMke with waight : And of late time, by Trochilike with Spring : without waight. All thefe, d.ij. by- lohn Dee his Mathematical! Preface . by Sunne or Sterres diredion ( in certaine time ) require ouerfight and reformati¬ on, according to the heauenly dsquinodiall Motion: befidesthe ineequalitie of their owne Operation . There remayneth (without parabolicall meaning herein) Afcrfctuall among the Philofophers,a more excellent, more commodious,and more maruei- lous way, then all thefe .* ofhauing the motion of the Primouant (or firft a?quino- diall motion, )by Nature and Arte,Imitated: which you fhall ( by furder fearch in waightier ftudyes ) hereafter, vnderftand more of. And fo, it is tyme to firiifh this Annotation,of Tymes diftindion,vfed in our common,and priuate affaires: The commoditie wherof,no man would want, -that can tell, how to beftow his tyme. Zograpllie^is an Arte Mathematicall, which teacheth and de- monftrateth , how , the Interfedlion of all vifuall Pyramides , made by any playne al signed, ( the Centre, diftance,and lightes,beyng de¬ termined) may be, bylynes,and due propre colours, reprefen ted. A notable Arte, is this*and would require a whole Volume, to declare the proper¬ ty thereof: and the Commodities enfuyng . Great fkili of Geometric, LsArithme- uke^PerJpeclme, and Antkropographie, with many other particular Artes,hath the Zo¬ grapher, nede of for his perfedion.For, the moft excellent Painter, (who is but the propre Mechanicien, & Imitator fenfible, of the Zographer) hath atteined to fuch perfedion,that Senfe of Man and beaft,haue iudged thinges painted, to be things naturalfand not artificial! :aliue, and not dead.This Mechanicall Zographer ('com¬ monly called the Painter)is meruailous in his fkill:and feemeth to haue a certaine diuine power: As,offrendes abfent,to make a frendly , prefent comfort : yea, and of frames dead, to giue a continual! , filent prefence : not onely with vs , but with our pofteritie, fomany Ages. Andfoprocedyng, Confider,How,in Winter, he can fhew you, the liuely vew of Sommers Toy, and riches:and in Sommer,exhibite the countenance of Winters dolefu.U State, and nakednes.Cities,Townes, Fortes, Woodes, Armyes, yea whole Kingdomes (be they neuerfofarre,or greate) can he,with eafbjbring with him, home (to any mans Judgement ) as Paternes liuely, of the thinges rehearfed. In one little houfe, can he,enclofe(with great pleafure of the beholders,) the portrayture liuely,of all vifible Creatures,either on earth,or in the earth, liuitig: or in the waters lying,Creping,fiyding,or fwimmingior ofany foule,or fly, in the ayre flying. Nay, in refped of theStarres,the Skie,the Cloudes: yea,in the fliew of the very light it felfe (that Diuine Creature ) can he match our eyes Iudgement,moft nerely. Whata thing is this'thinges notyet being,he can reprefent fo , as, at their being, the Picture fhall feame (in maner)to haue Created them. To what Artificer, is not Pidure,a great pleafure and Commodities Which of them all, will refufe the Diredion and ayde ofPidure'The Archited,the Gold- fmith'and the Arras Weauer : ofPidure,make great account. O ur liuely Herbals, our portraitures ofbirdes, beaftes,ancl fifties : and our curious Anatomies,which way,are they moll: perfedly made,or with moft pleafiire,ofvs beholden? Is it not, by Pidure onely? And if Pidure , by the Induftry of the Painter, be thus commo¬ dious and meruailous: what fhall be thought of Zographie, the Scholemafter ofPi- dure,and chief gouernor? Though I mention not Sculpture ,in my T able ofArtes Mathematicall : yet may all men perceiue,How,that Picture and Sculpture , are Si¬ fters germaine:and both, right profitable , in a Commo wealth.and of Sculpture, zf- well as of Pitiure,excellent Artificers haue written great bokes in commendation. Witnefle I take, of Georgia Vafarijittore Aretino:of Pomponius Gauricus : and other. To thefe two Artes, (with other, )is a certaine od Arte , called Althalmafat, much beholdyng: more, then the common Sculptor ,Entayler,Keruer, Cutter, Grauer, Foun - Iohn Dee his Mathematical! Preface. der, or Paynter(&c)knovf their Arte,to be commodious. Ar chltecture^to many may feme not worthy, or not mete, to be reckned -An obiettion, among the Artes UVtathematicall.To whom, I thinke good, to giue fome account of my fb doyng.N'ot worthy, (will they fay,)bycaufe it is but for building, of a houfe, Pallace, Church, Forte, or fuch like,groffe Workes.And you,alfo, defined the Artes OWathanaticalljLQ be fuch, as dealed with no Materiallor corruptible thing: and al- fo did demonftratiuely procedein their faculty ,by Number or Magnitude . Firft, you fee, that I covmt fetdyArchiteclurey among thole ^Artes Mathematically which The Aufwer, are Deriued from the Principals : and you know , that fuch,may deale with Na- turall thinges,and fenfible matter , Of which , fome draw nerer,to the Simple and abfolute Mathematicall Speculation, then other do . And though, the Architect ” procureth, enformeth, & diredeth,the Mechanicienf. o handworke, & the building ” aduall,ofhoufe,Caftell,or Pallace, and is ehiefludgeofthefame : yet , with him felfe (as chief CWafier and Architect > ) remaineth the Demonftratiue reafon and caufe, of the Mechaniciens worke.- in Lyne,plaine, and Solid : by Geometrically A- ” rithmeticall,Opticall,Muficall,AttronomicallyCofmographicall ( & to be brief) by all the ” former Deriued Artes Mathematically and other Naturall Artes,hable tot>e confir- ** med and ftablifhed.If this be fo: then, may you thinke, that Architecture, hath good and due allowance , in this honeft Company of Artes. CMathematicall Deriuatiue. I will,herein,craue Iudgement of two moft perfed Architefies : the one , being Vi- truuius, the Romaine : who did write ten bookes thereof, to the Emperour ^Augu¬ stus ( in whofe daies our Heauenly Archemafter, was borne ) : and the other, Leo Baptifla Alhertus , a Florentine : who alfo published ten bookes therof . oArchi- tetlura ( fayth Vitruuius ) est Scientia pluribus dijciplinis dr ruarijs eruditionibus or n at as cuius Iudicio probantur omnia , qua ab cater is Artifcibus perficiuntur opera . That is. Architecture, is a Science garnifhed with many doCtrines & diuerfe inftruCtions : by whofe Iudgement, all workes,by other workmen finifhed, are Illdged . It followeth.&z nafcitur ex Fabrica , dr Ratiocinatione.&e. Ratiocinatio auttm eft, qua,res fabrtcatas,S olertia ac ratione proportions >demonBr are atifi exphcare potest . jdrchiteffiure groweth ofFramingfind %eafoning.i?c. %ea* foningyis ibdifwnch of t hinges framed fiicith forecaft ,and proportion: can make demonstration^ and manif eft declaration . Againe. Cum , in omnibus enim re¬ bus, turn maxim l etiam in Architectura, hac duo infunt : quod fignifcatur, dr quod figni- ficat . Significant propofit a res, de qua dicitur rhanc autem Si^mficat Demonfiratioyrati- onibus doctrinarum explicata . Forafmuch as , in all t hinges* therefore chiefly in jdrc hit effur ey t hefe two thinges are :the thing fignifted: and that Oflicb fig • nifieth , The thing propounded , thereof Wjpedke, is the thing Signified. But ■Demonftr ation ,expreffed Toith thereafonsof diuerfe doSirines ftoth Jigni* fie the fame thing . After that .Ft liter aius fit , peritus Graphidos,eruditus Geometria , dr Optices non ignarus : infiructus Artthmetica-.hifiorias complures nouerit , Philofophos dmgenter.audiuerit:%Muficamftiuerit \ Medicina non fit ignarus, refionfa lurijperitoru nouerit: Afirologiam, Calife ratipn.es cognitas habeat .An Architect (fayth he) ought to ynderftand Languages yto be fkilfull ofF ainting , mellinftruffied in Geome * trie, not ignorant ofFerJpeffiiue , furnijhed'toith Arithmetike faue knowledge of many hiftories, and diligently haue hard Fhilofophers y haue f kill of Mu> fike, not ignorant ofFlyfike , know the aunfweres of Lawyers ,and haue Aftro - d.itj. nomiey lohn Dee his Mathematical! Preface. nomie, and the courfes C&lefliall , in good knowledge . He geueth reafon 5 or¬ derly, wherefore all thefe Artes,Do&rines,and Inftru&ions, are requisite in an ex¬ cellent Architect . And (for breuitie) omitting the Latin text, thus he hath. Secondly, it is behofefull for an Architedto haue the knowledge of Taint in?: that he may the more eafilie fajhion out fin patternes painted ,the forme of what Worke he liketh. And Geometrie,geuethto Architecture manyhelpes : and first teacheth the Vfe of the flute, and the Cumpafie: wherby (chiefly and eafilie ) the defcriptions of Buildinges , are defpatched in Groundplats: and the directions of S quires ,Leuells , and Lines. Likewifefiy TerJpeCtiue/he Light es of the be a* nen,are 1 Kell led Jin the buildinges : from certaine quarters of the World . (By Arithmetike ,the charges of Buildinges are fiummed together: the meafures are exprefied, and the hard questions of Symmetries, are by Geome tricall Meanes and Methods difcourjed on. tyre. Befides this ,of the Nature of tbinges( which in Greke is called ?u<7ioXo/i& ) Thilojophie doth make declaration . Which, it is neceffary ,for an Architect, with diligence to haue learned : becaufe it hath ma • nyand diners naturall queftions: asfpeciallyfin Aqueducdes . For in their courfes, leadinges about, in the leuell ground, and in the mountinges , the natu * rail Spirit es or breathes areingendred diners Wayes : The hindrances , which they caufe, no man can helpe, but be, 'Which out ofThilofophie, hath learned the originall caufes of thinges . Like wife , who foeuer flail read Ctefibius, or Ar» chimedes bookesf and of others, who haue written fuel? flules)can not thinke,as they do : 'bnleffe he f ull haue receaued of Thilofophers , inflruCiions in thefe thinges . And Mufike he mufti nedes know : that he may haue lender standing, both of Tegular and Mathematicall Mufike: that he may temper Well his Ba* liftes, Catapultes , and Scorpions. iyc. Moreouer ,the Brafen Veffels , which in T beatres,ate placed by Mathematicall order, in ambries fender the Steppes: and the diuerfities of the foundes (which y Grecians call w ) are ordred according to Mu f call Symphonies W Harmonies:being diflributed in y Circuites, by (Di* ateffaronJDiapente,and (Diapafon. That the conuenient boy ce, of the players found, whe it came to thefe preparations, made in order , there being increafed: Withy increafing, might come more cleare & pleafant,to y eares of the lokers on. i&c.And ofAJlronomieJs knoweyEaSt, Weft, Southland North . The fajhion of the heauen , the AEquinox , the Solfticie , and the courfe of the fibres. Which thinges grnleafl one know:he can not perceiue ,any thyng at all, the reafon of Ho* rologies.Seyng ther fore this ample Science ,is garnified , beautified and flored. With fo many and fundry f kils and knowledges:! thinke , that none can iuflly ac* count themfidues ArchiteCfes ,of the fnddeyne. But they onely,who from their childes yeares ,afcendyng by thefe degrees of knowledges, beyngfoUered bp with the atteynyng of many Languages and Artes , haue Wonne to the high T aber* nacle of Archiffiure.is'c.And to whom Nature hath giuen fuch quicke Circum* fieCfionjbarpnes of Witt, and Memorie,that they may be bery abfoktelyfkilU fullin Geometrie , . Afironomie , Mufike, and the reft of the Artes Mathemati * John Dee his MatliematicallPrrface, calkSticb fur mount and pajfe the callyng3andftatt', of Ar chit e tides: and are he* A t&fatbe* come Mathematiciens.e&c.Mndthey are found feldome. As, in tymes paft } 1 \>as matieien, Mriftarchus*Samius:Tbildlaus3andArcbytasfTarentynes: Apollonius Bergpus: Eratofthenes Cyreneus: Archimedes 3and SeopasJSyracufians. Who alfojeft to theyr pofteritiepnany Engines andGnomomcaUypork.es: by numbers andnatu* rail meanes ftnuented and declared. Thus much., and the fame wordes (in fenfejin one onely Chapter of this Inco¬ parable ArchitectVitruutus,{CdX\ youfinde.And if you (hould , hut take his boke-in your hand,and (lightly loke thorough it, you would fay ftraight way : This is Geo- yitrmws* metrie,Anthmetike, ^Astronomic, Mufike,Anthropographie,fJjdragogie, Horometrie.&c, and (to coclude ) the Storehoufe of all workmafhip . No w,let vs liften to our other Iudge,our FlorcntmeyLeo.Baptifta:a.nd narrowly confider,how he doth determine of Architecture. Sed anteoy vltra progrediar.&c. But before Iprocede any further ((ayth he)/ thinkefthat I ought to exprejfe 7 yohat man I leould haue to bee al* lowed an Architect. For ft ypill not bryng in place a Carpenter : as thoughyou might font. pare him to the Chief M afters of other jtirtes. For the hand of the • farpenterfs the Ar chit e tides Inftrument, But I ypill appoint the Architect to be that manfttoho hath the J kill ft by a certaine and meruailous meanes and 1 Vayft n tu ' both, in mindeand Imagination to determine: and alfo in yporke to finifh : yphat ” yoorkes foeuerfty motion of ypaight}and cupplingand framyng together of bo* dyespnay moH aptly be Qjmtnodious for the yportbieft Vjes of Man, And that he » may be able toperforme thefe thinges } he hath nede of atteynyng and knowledge of the beft3and moft yporthy thynges. tyre . 1 he yphole Feate of Architecture in buildy ngponfUeth in Line ament es 3and in Framyng . ylna the yphole power; and fktll ofLineamentesftendetb to this : that the right and ahfolute Ip ay may he had3of Coaptyngand ioyning Lines and angles'.by yohich 3the face of the buiti dyng or frame pnay be comprehended and concluded , And it is the property of Lineamentes pto prefcribelmto buildynges }and euery part of tbem3an apt place , CT certaine ntiber : a Worthy maner}and afemely order : thatfoft yphole forme and figure of the buildyngpnay reft in the 1?ery Lineament es. (type . And ype may * The Im- preferibe inmynde and imagination the ypbole formes ft all materiall ftuffe be* matcrialitl" Jtig ft eluded Which point ype f hall atteynefty Fbotyng and forepointyng the an* glespnd lines fty a Jure and certaine diretiiion and connexion. Seyng thenfthefe thinges y are thus : Lineamente 3fhalbe the certaine and constant hrefcribyng3 WbatftAnu- concerned in mynde: made m lines and angks:and finiftbed ypith a learned minde mntts' ' andypyt. We thanke you Maher Baptist, that you haue fo aptly brought your „ Arte , and phrafe therof , to haue (ome Mathematical! perfection : by certaine or- ,, <3S(jte, der, nuber, forme, figure, and Symmetric mentall: all naturall & fenfible ftuffe fet a » part. N ow, then, it is euidcnt,(Gentle reader)how aptely and worthely , I haue preferred Architecture, to be bred andfoftered vp in the Dominion of the pereles Princefe, (JUathematicx : and to be a naturall SubieCt of hers . And the name of Architecture, is of the principalitie, which this Science hath, aboue all other Artes. And Plato affirmeth , the Architect to be AH after ouer all, that make any worke. Wherupon,he is neither Smith,nor Builder: nor3(eparatcly, any Artificer: but the d.iiij. Hed, Anno. IJ5P, lohnDee his Mathematical! Preface. Hed,thcProuoft, the Dirc£ter,and Iudge of all Artificial! workes, and all Artifi- cers.For,thc true Architects is hable to teach ,D em onft rate,diftribu te,dcfcrib c , and Iudge all workes wrought. And he,onely,Iearcheth out the caufes and reafons of all Artificial! thynges.Thus excellent, is Architecture : tho ugh few(in our dayes)at- teyne thereto : yet may not the Arte, be otherwife thought on, then in veiy dede it is worthy. Nor we may not, of auncient Artes,make new and impeded Definiti¬ ons in our dayesrfor fcarfitie of Artificers : No more, than we may pynche in, the Definitions of Wifedomc, or Honefite , or of Frendejhyp or of Iujlice . No more will I confent,to Diminifli any whit, of the perfedionand dignitie , ("by iuft caufe ) al¬ lowed to abfolute Architecture. V nder the Diredion of this Arte , are thre prin- cipall,necdTary Mechanic all Artes . Namely , Horvfwg , Fortification , and T{aupegie. Horvfingj I vnderftand,both for Diuine Seruice,and Mans common vfage: publike, and priuate.Of Fortification an dNaupegie, ftraunge matter might be told you: But percnaunce/ome will be tyred, with this Bederoll, all ready rehearfcd: and other fomc, will nycely nip my grofle and homely difcourfing with you : made in poll: haft : for fearc you fliould wante this true and frcndly warny ng, and taft giuyng, of the Power JSIathematicall . Lyfe is ihort, and vncertaine : Tymes are periloufe; Bcc . And ftill the Printer awayting, for my pen ftaying : All thefe thingcs,with farder matter of I-ngratefulnes, giue me occafion to paffe away , to the other Artes remainyng, with all fpede pofsible. Xhc Arte of Nauigation, demoiiftrateth how, by the fhorteft good way, by the apteft Diredtio,& in the fhorteft time, a fufficient Ship,betwene any two places (in paffage Nauigable,) afsigned : may be codinded: and in all ftormes,& naturall difturbances chauncyng, how, to vie the beft pofsible meanes , whereby to recouer the place fir ft afsigned . What nede , the Mafler Pilote , hath of other Artes , here before recited, it is eafie to knowras, of Hydrographies AChonomky^ACirologie , and Horome- trie . Prefuppofing continually, the common Bafe,andfoundacion or all: namely Arithmetike and Geometric. So that,he be hable to vndcrftand,and Iudge his own neceflary Inftrumehtes,and furniture N eceftary: Whether they be pcrfedly made or norand alfo can, (if nede be) make them,hym fclfe. As Quadrantes, The Aftro- nomers Ryng,The Aftronomers ftaffe,The Aftrolabc vniucrfaii. An Hydrogra- phicall Globe.Charts Hydrographicall,true,(not with parallell Meridians) . The Common Sea CompasrThe Compas of variacion: The Proportionalfand Para- doxall Compafle$(of melnuented,forourtwo Mofcouy Mafter Pilotes,atthc re- queft of the Company) Clockes withfpryng: houre,halfe houre,and three houre Sandglalfes: & fundry other Inftrumetes: And alfo, be hable,on Globe, or Playne todeferibe the Paradoxall Compalfe : andduely to vie the fame,to allmanerof purpofes, whereto it was inuented. And alfo, be hable to Calculate the Planetes places for all tymes. ,c . T Moreouer,with Sonne Monc orSterrefor without)be hable to define the Lon- gitude Be Latitude of the place, which he is in: So that,the Longitude & Latitude of the place,from which he fayled,be giuen : or by him, be knowne. whereto, apper- tayneth expert meanes, to be certified euer,of the Ships way . &c. And by toreie- ing the Rifing,Settyng , N oneftedyng , or Midnightyng of cer taine tempeftuous fixed Sterres : or their Coniun<ftions , and Anglynges with the Pranetes , &c.he ought to haue expert coniedure of Stormes , T empeftes , and Spoutes : and iuch lyke Meteorological! effe<ftes,daungerous on Sea. For (as Plato fayth y)Mutanonest lohn Dee his Mathematical! Preface. epportunitatefy tempo rum prefentire, non minus rei militari, quam Agriculture? Nauiga - tionitfc conuenit. T o forefee the alterations and opportunities of tymesjs come * nient , no lefe to the Art of JVarre } then to HuJ bandry and Navigation. And befides fuch cunnyng meanes , more euident tokens in Sonne and Mone , ought of hym to be knowen: fuch as(the Philofophicall Po ex€)Virgilius teacheth, in hys GeorgikesAN'ncxc he fayth, Sol quofy dr exoriens dr quum fe condet in vndas, Signa dabitySolem certifima ftgna fequuntur.drc . - - — 'Horn ftpe videmns , Ipjtus in vultu varios err are colores. Caruleus, pluuiam denunciat dgneus Euros. Sin macule incipient rutilo immifcerier ign ’t , Omnia turn par iter vento^nimbifq, videbis Eeruere: non ilia quifquam me nocie per altum Ire , necp a terra moueat conuellere funem . dfc. Sol tibi ftgna dabit. Salem quis dicerefalfum lAudeati - - drc. And fo of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and Bcaftes, and of many thynges els,a certaine Sympathicall forewarnyng may be had: fome- tymes to great pleafure and proflit , both on Sea and Land. Sufnciendy, for my prefent purpofe , it doth appeare, by the premiffes , how Mathematically the Arte of Navigation, is:and how it nedeth and alfo vfeth other Mathematicall Artes : And now, if I would go about to fpeake of the manifold Commodities, commyng to this Land, and others, by Shypps and Navigation , you might thinke , that I catch at occafions , to vfe many wordes , where no nedc is. Y et, this one thyng may I, (iuftly) fay. In Nauigaiion, none ought to haue grea¬ ter care, to be fkillfull,then ourEngiiih Pylotes. And perchaunce,Some, would more attempt: And other Some,more willingly would be aydyng, if they wift cer- tainely,What Priuiledge,God had endued this Hand wirh,by reafon of Situation, moft commodious for Nanigation, to Places moft Famous 8c Riche. And though, (of* Late) a young Gentleman,a Courragious Capitaine , was in a great ready- nes, with good hope, and great caufes ofperfuafion,to haue ventured, for a Dll- CO lierye, (either Wejlerly, by Cape de Paramantia : or , aboue Noua Zemla, and the Cjremijfes)and was, at the very nere tyme of Attemptyng , called and em¬ ployed otherwife(both then, and fince,)in great good feruice to his Countrey , as the Irifh Rebels haue * tailed: Yet, I fay, ( though the fame Gentleman , doo not hereafter, deale therewith)Some one, or other ,fhould liften to the Matter: and by good aduife,and diferete Circumfpe&ion , by little, and little, wynne to the fuffn cient knowledge ofthatTradeand Voyage: Which, now, I would be Tory, (through CarelefnefTe,want of Skill, and Courtage, ) fhould remayne Vnknowne and vnheard of. Seyng, alfo,we are herein, halfe Challenged, by the learned,by halfe requeft,publifhed. Therof,verely, might grow Commoditye , to this Land chiefly, and to the reft of the Chriften Common wealth, farre pafling all riches and worldly Threafure. Thaumaturglfee,is that Art Mathematicall, which giueth cer¬ taine order to makeftraunge workes , of the fenfe to be perceiued, and of men greatly to be wondred at. By fundiy meanes,this Wonder- worke is wrought. Some,by Fneumatithmie . As the workes of Ctefibius and Hero , A.j. Some Georgia, i. *Anno.\%6j s.H.g . *Anno.i$6g lohtt Dee liis Mathematical! Preface. Soineby waight.wherof T imam lpeaketh.Some,by Strirtges ftrayned,or Springs, therwith Imitating liucly Motions.Some, by other meanes,as the Images of Mer¬ curic .-and the brafen hed,fnade by Albertus Magnus, which dyd feme tp Ipeake.-Btn?- thins was excellent in thefe feates. T o w h o m ,CaJsi od or us writyng,fayth.2 our put* pofe is to know profound thyngesiand to jkew meruajles. <By the diSpofition of jour Arte , Metals do low : (Diomedes ofbrafie , doth blow a Trumpet loude : a brafen Serpent hiffetkbyrdes made ,fingfwetelj. Small thynges lee rehearfe ofyou,loho can Imitate the heauen.ejrc. Of the ftraunge Selfmouyng, which, at *4mo. 1551 Saint Denys , by Paris ,*Ifaw , ones or twife ( Orontius beyng then with me, in C ompany jit were to ftraunge to tell. But fome haue written it. And yet, (I hope) it is there, of other to be fene. And by PerJpecJiue alio ftraunge thingcs,arc done. As partly (before) I gaue you to vnderftandin Perjpetfm. As, to fee in the Ayre, a loft, thelyuely Image of an other man , either walkyng to and fro : or ftandyng ftill. Likewife, to come into anhoufe,and there to fee the liucly ftiewofGold,Siluer or precious ftones:and commyng to take them in your hand , to finde nought but Ayre. Hereby, haue fome men ( in all other matters counted Wife ) fouly ouerfnot the fellies : mifdeaming of the meanes.Thaforeiayd CUudmCAeflinus. Hodie mag- na literature viros & magne refutdtioms ‘videmus , opera qua dam quafi miranda ,fupra Nat ura putare: de qmbnsin PerjfeBiua dobhis caufamfdciliter reddidijfit.Thzt is JSlow a dayes ,!»e fee fome men, yea of great learnyng and reputation, to Iudge certain 1 V0rk.es as meruaylous ,ahone the power of Mature : Of lefich 1 vorkes,one that leerefkillfull in Terfpeltiue might eafely hauegiuen the faufe. Of Archimedes Sphere, Cicero witneftqth .Which is very ftraunge to tftinke on. For when Archie medes{£ ayth he) did faflen in aSph cere , the mouynges of the Sonne, Mone,and of the fine other r?lanets,he did, as the God,H>hiclfin Timeeus of? Into) did make the leorld. That ^one turnytig,Jhould rule motions mo filmlike in flownes, arid fw fines. But a greater caufe of meruayling we haue by CUudianm report hereof. Who aiftrmeth this Archimedes worfo, to haue ben of Glafte. And difeourfeth of it more at large: which I omit. The Doue ofwood , which the Mathematicien Ar¬ ch jt as did make to fly e, is by iMgellius ipoken of.O tDadalus ftraunge Images, Plato reporteth . Homer e of Vulcans Selfmowtrs , (by fccret whcles)lcaueth in writyng . Ari- fiotkja hys Volmkes, of both, makethmention. Meruaylous was the workeman- -£ftyp,of late dayes, performed by good f kill otTrochilike. dec • Tor in Noremberge, A flye oflern, beyng let out of the Artificers hand,did(as it were)fly about by the geftes,at the table, and at length, as though it were weary , rctoutne to his mafters hand agayne . Moreouer, an Artificial! Egle , was ordred , to fly out of the fame Towne,a mighty way, and that a loft in the Ayre, toward the Empcroiir comming thethenand followed hym, beyng come to the gate of the townc.* Thus,you fee, What,ArteMathematicallcan perform e,when Sldll , will, Induftry, andHabili- ty, are duely apply ed to profe. A DC ef'o't An d for thefe, and fuch like marueilous Ades and Feates, Naturally ,Mathc- ApologeJca/L matically,and Mechanically, wrought and contriued : ought any honeft Student, and Modeft Chriftian Philofopher,be countcd,& called a Coniurer ? Shall the folly of Idiotes, and the Mallice of the Scornful!, to much preualle, that He, who feeketh no worldly game or glory at their handes : But onely,of God, the threafor of heauenly wifedome,& knowledge of pure veritie : Shall he (I fay) in the meane SDe his qua Munch mi- rabiltter eue- niunt. cap. 8 Thfc. i. ST lohii Dee his Mathematical! Preface* fpace, be robbed and fpoiled of his honed: name and fame < He that feketh ( by Si Paules aduertifement ) in the Creatures Properties , and wonderful! vertues, to finde iufte caufc, to glorifie the Asternalfand Almightie Creator by : Shall that man, be ( in hugger mugger ) condemned, as a Companion of the Helhoundes , and a Galley and Coniurer of wicked and damned Spiritesc’ He that bewaileth his great want of time,fufficient(to his contentation)for learning of Godly wifdome, and Godly Verities in : and onely therin fetteth all his delight : Will that ma leefe and abufe his time, in dealing with the Ghiefe enemie of Chrift our Redemer: the deadly foe of all mankinde : the fubtile and impudent peruerter of Godly Veritie: the Hypocriticall Crocodile: the Enuious Bafilifke, continually defirous, in the twinkeofaneye,todeftroy all Mankinde, both in Body and Soule , eternally i . Surely (for my part, fomewhat to fay herein) I haue not learned to makefobrutifh, and fo wicked a Bargaine . Should' I, for my xx.or xxv. yeares Studie : for two or three thoufand Markes fpending : feuen or eight thoufand Miles going and- trauai- ling, onely for good learninges lake r And that, in all maner of wethers : in all ma- ner of waies and paftages : both early and late : in daunger ofviolence by man : in daunger o.fdcftru&ion by wilde beaftes : in hunger : in third: : in perilous heates by day, with toyle on foote : in daungerous dampes of colde,by night, almoft be- reuing life : (as God knoweth) : with iodginges, oft times,to iinall eafe : and font- , time to Idle fecuritie. And for much more (then all this) done & fuifred, for Lear¬ ning and attaining of Wifedome : Should I ( I pray you) for all this,no otherwife, nor more warily : or (by Gods mercifulnes ) no more luckily, haue fiftied, with fo large, and coftly,a N ette, fo long time in drawing (and that with the helpe and ad- uife of Lady Philofophie,& Queen e Theologic) : but at length, to haue catched, and drawen vp,* a Frog i Nay, a Deuill < For,fo,doth the Common peuifh Pratler * ^ prouerfa Imagine and Iangle: And,fo,doth the Malicious fkorner,fecretly wiftie,& brauely FaJfefi[bt, and boldly face down, behinde my backe . Ah, what a miserable thing, is this kinde ^caught of Men < How great is the blindnes & boldnes,of the Multitude, in thinges aboue re&% their Capacitie < What a Land: whata People : what Man ers : what Times are thefe < Are they become Deuils,them felues: and, by falfe witnelfe bearing againft their Neighbour, would they alfo, become Murderers * Doth God, fo long geue- them refpite, to reclaime them felues in, from this horrible {laundering of the gilt- leife : contrary to their owne Confciences : and yet will they notceafe < Doth the Innocent,forbeare the calling of them, Juridically to aunfwerehim,accordingto the rigour of the Lawes : and will they defpife his Charitable pacience 1 As they, aeainft him, by name, do forge, fable, rage, and raife {launder, by Worde & Print: Will they prouoke him, by worde and Print, likewife, to N ote their Names to the World : with their particular deuifes, fables, beaftly Imaginations, andvnchriften- like {launders c’ Well : Well . O (you fuch ) my vnkinde Countrey men . O vn- naturall Countrey men . O vnthankfull Countrey men . O Brainficke, Raflie, Spitefull,and Difdainfull Coun trey men . Why opprefteyou me, thus violently, with yo ur {laundering of me : Contrary to Veritie: and contrary to your owne Confciences < And I, to this hower, neither by worde, deede, or thought, haue ben e, any wav, hurtfull, damageable, or iniurious to you,or yours < Haue I,fo long, fo dearly, fo farre,fo carefully, fo painfully ,fo daungcroufly fought 6c trauailed for fhelearnlng of Wifedome,& atteyning ofVertue : And in the end(ln youriudge- met)am I become, worfe, then when I begaf Worfe,the a Mad man.? A dangerous Memberin the Common Wealth: and no Member of the Church of Chrift? Call you this, to be Learned ? Call you this, to be a Philofopher ? and a louer of Wife- dome ? To forfake the ftraight heauenly Way *. and to wallow in the broad way of A.ij. dam- Iohn Dee his Mathematicall Preface . damnation ? To forfake the light of heauenly Wifedome: and to luike in the dun¬ geon of thePrinee of darken ene ? To forfake the Veritie of God,&his Creatures: and tofawne vpon the Impudent, Craftic,Obftinate Tier, and continuall difgracer of Gods V eritie, to the vttermoft of his power l T o forfake the Life & Bliffe Aiter- nall: andto cieauevntothe Author of Death euerlafting ? that Murderous Tv- rant, moft gredily awaiting the Pray of Mans Soule ? Well : I thanke God and ourLordelefus Chrift, for the Comfort which I haue by the Examples of other men, before my time : To whom, neither in godlines of life, nor in perfection of learning, I am worthy to be compared : and yet, they fuftained the very like Iniu- ries , that I do : or rather, greater. Pacient Socrates , his ^Apologie will teflifie : Apu- leius his ^Apologies, will declare the Brutifhneffe ofthe Multitude . loannes Via*. r, Earle of Mirandula, his Apokgie will teach you, of the Raging flaunder ofthe Ma¬ licious Ignorant againft him . Joannes T rithemim, his Apologie will fpecifie, how he had occafion to make publike Proteftation : as well by reafon of the Rude Sim¬ ple : as alfo,in refpeft of fuch,as were counted to be of the wifeftfort of men. Ma- „ ny could I recite : But I deferre the precife and determined handling of this mat- O** „ ter: being loth to deted the Folly & Mallice of my N atiue Countrey men.*Who, „ fo hardly, can difgeft or like any extraordinary courfe of Philofophicall Studies: ,, not falling within the Cumpaffe of their Capacitie : or where they are not made priuie of the true and fecrete caufe, of fuch wonderfull Philofophicall Feates. Thefe men, are of fower fortes, chiefly . The firft, I may name, Faine pratling bu- fte bodies .-The fecond , Fond Frendes : The third, Imperfectly serious: and the fourth, t Malicious Ignorant . To cche of thefe (briefly, and in charitie ) I will fay a word I# ortwo,andforeturne to my Preface. Vaine pratling bufe bodies ,vfe your idle allemblies,and conferences, otherwife, then in talke of matter, either aboueyour Capacities, for hardnefle : or contrary to your Confciences, in Veritie . Fonde 2 . Frendes, leaue of,fo to commend your vnacquainted frend,vpon blindeaTedion: As, becaufe he knoweth more, then the common Student: that, therfore, he inuft needes be fkiifull,andadoer,infuch matterandmaner,asyouterme Coniuring. Wecning,thereby, you aduaunce his feme : and that you make other men, great marueilers of your hap, to haue fuch a learned frend . Ceafe to aferibe Impietie, where you pretend Amitie . For, if your tounges were true, then were thatyour frend, Vntrue, both to God, and his Soucraigne . Such Frendes and Fondlinges, I fliake of, and renounceyou : Shakeyou of, your Folly. Imperfectly serious, to you, 3* do I fey: that (perhaps) well, do you Meane : But farre you mifle the Marke : If a Lambe you will kill,tofeede the flocke with his bloud. Sheepe, with Lambes bloud, haue no naturall fuftenaunce : No more, is Chriftes flocke, with horrible flaunders, duely edified. Nor your feire pretenfe, byfuch rafhe ragged Rheto- rike,any whit,well graced. But fuch,asfo vfe me, will finde a fowle Gracke in their Credite . Speake that you know : And know, as you ought : Know not, by Heare Eiy, when life lieth in daunger. Search to the quicke,& let Charitie be your guide. 4. c Malicious Ignorant , what (hall I fey to thee ? Prohibe linguam tuam a malo . ^Ade - tr action c parcite lingua: . Caufe thy toung to refraine fro euill. fefraineyour toung from flaunder . Though your tounges be fliarpned. Serpent like, & Adders poy- Pfai, 1 40 . fon lye in your lippes : yet take heede,and thinke, betimes, with your felfe. Fir lin¬ gua fm non flabilietur in terra . Firum violentum venabitur malum , donee pracipitetur. For,fure I am, fhtja faciet Dominus Iudicium afflicti .• & vindictam pauperum . Thus, I require you, my aflured frendes, and Countrey men ( you Mathemati- dens,Mechaniciens,andPhilofophers,Charitable and diferete) to deale in my behalfe. lohn Dee his Matliematicail Preface. behalf, with the light & vntrue toungcd, my enuious Aduerfaries,or Fond fr ends » And farther, I would wifhc, that at leyfor, you would confider,how Bafdius Mag* nus, layeth CMofes and Daniel:, before the eyes of thofe, which count all fuch Stu¬ dies Philofophicall (as mine hath bene) to be vngodly , or vnprofitable . Waye well S. Stephen his witnefle of CMofes . Erudttus est CMofes omni Sapientta JUgyptioru: ^ £ & eratpotens in verbis & openbm fuis . Mofes leas inRmfifed in all rnaner of^ife* dome of the ^Egyptians : and he "ivas of power both in his hordes ? and forties. You lee this Philofophicall Power & Wifcdome, which CMofes had, to be nothing mifliked of the Holy Ghoft. Yet Plinius hath recorded, Mofes to be a wicked Magi* cien . And that (of force) muft be, either for this Philofophicall wifedome,iearned, before his calling to the leading of the Children of Jfrael : or for thofe his won- ders,wrought before King Pharao , after he had the conducing of the Ifraelites. As concerning the frft,you perceaue, how S. Stephen, at his Martyrdome ( being full of the Holy Ghoft) in his Recapitulation of the olde Teftament,hath made men¬ tion of Mofes Philofophie : with good liking of it : And Baflius Magnus alfo, auou- cheth it, to haue bene to Mofes profitable ( and therefore, I fay, to the Church of God, neceflary). But as cocerning Mofes wonders, done before King PharaovGod , him felfe, fayd : Vide vt omnia osfenta, qua pofui in manu tua , facias coram Pharaowe. See that thou do all thofe bonders before Pharao ? Tvhicb I haue put in thy hand. Thus, you euidently perceaue, how vaShly^P limits hath llaundered Mofes, ofvayne Lib. 30. fraudulent CMagike, faying : EH Sr alia Magices F actio, a CMofe , Iamne,& Iotape, Iii* Cap, 1. dais pendens : fed multis millibus annorum pojl Zoroastrem.frc. Let all ftich , there¬ fore, who, in Iudgement and Skill of Philofophie, are farre Inferior to Plinie, take good heede,leaft they ouerfhoote them felucs rafhly , in Iudging of Phtlofophcrs Hraunge Actes • andtheMeanes,how they are done . But, much more, ought they to beware of forging, deuifing, and imagining monftrous feates, and wonderfull Workes, when and where, no fuch were done : no, not any fparke or likelihode,of fuch, as they, without all fhame, do report . And ( to conclude ) moft of all, let thembea!hamedofMan,andafraide of the dreadfullandlufteludge: bothFo- iiflily or Malicioufly to deuife : and then,deuililhly to father their new fond Mon- fters on me : Innocent, in hand and hart : for trefpacing either againft the lawe of God, or Man, in any my Studies orExercifes, Philofophicall, or Mathematical!: As in due time, I hope, will be more manifeft. No w end I, with ArcllCIHclftriC. Which name, is not fo new, as this Arte is rare.For an other Arte,vnder this, a degree(for fkill and power) hath bene indued with this Englifh name before. And yet, this, may feme for our purpofc, fufficiendy,at this prefent. This Arte, tgacWth to bryng to adtuall ex¬ perience fenhble,all worthy conclufions by all the Artes Mathema¬ tical! purpofed, & by true Naturall Philofophie concluded :& both addeth to them a farder fc :ope,in the termes of the fame Artes , & al¬ io by,hys propre Method, and in peculier termes, procedeth , with helpe of the forefayd Artes , to the performance of complet Expe- rieces, which of no particular Art, are hable (Formally) to be challen¬ ged . If you remember, how we considered ^Architecture, in refpetft of all com¬ mon handworkes : fame light may you haue, therby, to vnderftand the Souerain- ty and propertie of this Science. Science I may call it,rather, then an Arte: for the excellency and Mafterlhyp it hath , ouerfomany , and fo mighty Artes and A.iij. Sciences. 1. 2. lohn Dee his Mathematicall Preface. Sciences. And bycaufe it procedeth by Experiences ,and fearcheth forth the caufes of Conclufions,by Experiences : and alfo putteth the Conditions them felues, in Experience ,it is named of fom c^Scientia Experimental^ . The Experimentall Sci* ence. Nicolaus Cufanus termeth it fo, in hy s Experimentes Statikall , And an other Pbilefopher , of this land Natiue ( the flcure of whofe worthy fame, can neuer dye nor wither) did write therof largely, at the requeft of Clementthe fixt. The Arte carrieth with it, a wonderfull Credit : By reafon, it certefieth , fenfibly, fully, and completely to the vtmoft power of Nature, and Arte. This Arte,certifieth by Ex¬ perience complete and abfolute : and other Artes,with their Argumentes,and De- monftrations , perfuaderand in wordes,proue very well their Concluflons. *. But U* wordes,and Argumentes,are no fenfible certifying.- nor the full and finall frute of Sciences praftifable. And though fome Artes,haue in them, Experiences, yet they are not complete , and brought to the vttermofl, they may be ftretched vnto,and applyed fenfibly. As for exam pie: the Naturall Philofopher difputeth and malceth goodly fliew of: reafon : And the Aftronomer,and the Optical! Mechanicien,put fome thynges in Experience: but neither, all, that they may: nor yet fufficiently, and to the vtmoft,thofe,which they do. There, then,the Archemajler ftcppeth in, and leadeth forth on , the Experiences , by order of his dodrine Experimentall , to the chief and finall power of Naturall and Mathematicall Artes.Oftwo or three men, in whom, this Defcription of ArchemaBry was Experimentally rifled, I haue read and hardrand good record, is of their fuch perre&ion.. So that,this Art, is no fan- tafticall Imagination: as fomeSophifter, might, Cum fats Infolubiltbtu^ make a flo- rifh: and dalfell your Imagination: and dafln your honeffc defire and Courage, from beleuing thefe thinges,fo vnheard of,fo meruaylous,& of fuch Importance. Well: as you will.I haue forewarned you.I haue done the part of a frende.-I haue dischar¬ ged my Duety toward God:for my finall Talent, at hys moft mercyfull handes re- ceiued. T o this Science,doth the Science Alnirangiat, great Seruice. Mule nothyng of this name. I chaunge not the name, fo vfed, and in Print publifhed by other: beyng a name, propre to the Science. Vnderthis,commeth <^Ars Smtrillia , by Artefbius, briefly written . But the chief Science , of the Archemafter , ( in this world)as yetknowen , is an other ( as it were) OPTICAL Science : wherof, the name finall be toldf God willyng) when I finall haue fome, ( more iuftjoccafion, therof, to Dilcourfe. Here,I muft end , thus abruptly ( Gentle frende, and vnfayned louer of honeft and neceflary verities.) For,they,who haue(for your fake, and vertues caufe)re- quefted me,(an old forworn e Mathematicien) to take pen in hand : ( through the confidence they repofed in my long experience: and tryed fincerity) for the decla- ryng andreportyng fomewhat,of thefruteand commodity, by the Artes Ma¬ thematicall, to beatteynedvnto:euenthey, Sore agaynft their willes,are forced,for fundty caufes, to fatiffie the workemans requefl: , in endyng forthwith: He, fo feareth this, fo new an attempt,& fo coftly: And in matter fo flenderly (he- therto)amongthe common Sorte ofStudentes,confideredor efiemed. And where I was willed, fomewhat to alledge, why, in our vulgare Speche,this part of the Principall Science of Geometric, called Euclides Geometricall Elementes, is publifhed, to your handlyng : being vnlatined people, and not Vniuerfitie Scholers : Verily,IthinkeitnedelefTe. , • i* For, the Honour,and Eftimation of the Vniuerfities,and Graduates, is, hereby, nothing diminiflied. Seing, from, and by their Nurfe Children, you ' receaue all this Benefite : how great foeuer it be. Neither lohn Dee his Mathematical! Preface. N either are their Studies, hereby, any whit hindred. No more, then thxltalian a , Vniuerfities, 2S Academia Bone mentis, Ferrarienfs, Florentine, Medielanmfis, Patauina, Pagienfis, Pentfma, PifanayRomamfienenfis, or any one of them, finde them fellies, any deale, difgraced, or their St udies any thing hindred , by F rater Lucas de Burge > or by ‘liicelaus T artalea, who in vulgar Italian language, haiie publifhed, not onely * Euchdes Geometric, but of Archimedes feme\vhat : and in Arithmetike and Practical!. Geometric, very large volumes, all in their vulgar fpeche . N or in Germany haue the famous Vniuerfities, any tiling bene diicontent with Albertns Durerus, his Geo¬ metrical! Inftitutions in Dutch; orwith Gulklmus Xy lander, his learned tranflation of the firftfixe bookes oPEuclide, outofthe Greke into the high Dutch .Nor with : Gmlterm H . Riffius , his Geometricall Volume : very diligently tranflated into the high Dutch tounge, and publifhed . Nor yet the Vniuerfities of Spaine, or Portu- gall, thinke their reputation to be decayed ; or fuppofe any their Studies to be hin- - dred by the Excellent P. ?{onmus, his Mathematical! workes, in vulgare fpeche by him put forth . Haue you not, likewife, in the French tounge, the whole Mathe¬ matical! Quadriuie ? and yet neither Paris, Orleance, or any of the other Vniuer¬ fities ofFraunce, at any time, with the T ranflaters,or P ublifhers offended : or any mans Studie thereby hindreeb And furely , the Common and V ulgar Scholer ( much more, the Gramarian) 3 . before his comming to the Vniuerfitie, ihall ( or may) be , now (according to Plato his Counfell) fuiriciently inftruded in Arithmetike and Geometrie, For- the better and eafier learning ofallmaner of P hilofophie, Academically P erf ateticall. And by that meanes, goe more cherefully, more fkilfully, and fpedily fonvarde, in his Studies, there to be learned. And,fo, in leffe time,profite more,then (otherwife) he fliotild, or could do. J ••• Alfo many good and pregnant Engliihe wittes, of young Gen tlemen, and of 4« other, who neuer intend to meddle with the profound fearchand Studie of Philo- fophie ( in the Vniuerfities t obe learned ) may neuertheleffe, now, with more eafe and libertie, haue good occaiion , vertuoufly to occupie the fliarpheffe of their wittes : where,els (perchance ) otherwife, they would in fond exercifes,fpend ( or rather leefe) their time : neither feruing God : nor furderingthe Weale, common orpriuate. And great Comfort, with good hope, may the Vniuerfities haue, by reafon of 5 . this Engltfe Geometrie,. and Mathematicall Preface, that they (hereafter) Ihall be the more regarded , efteemed , and reforted vnto. For, when it ihall be knowen and reported, that of the Mathematicall Sciences onely, fuch great Commo¬ dities are enfuing ( as I hauefpeciiied ) : and that in dede, fome of you vnlatined Studentes, can be good witneile, of fuch rare fruite by you enioyed (thereby) : as either,before this, was not heard of; or els,n0tfb fully credited: Well,may all men conie&ure, that farre greater ayde,and better furniture, to winne to the Perfection „ ofall Philofophie,may in the Vniuerfities be had: being the Storehoufes & Threa- Vniuerfities lory of all Sciences, and all Artes, neceflaryfor the beft, and moft noble State of »» Common Wealthes. , „ Befides this, how many a Common Artificer, is there, in thefe Reaimes of 6. England and Ireland, that dealeth with Numbers, Rule, & CumpafTe : Who with their owne Skill and experience, already had, will be hable ( by thefe p-00d helpes and informations) to finde out, and deuifc,new workes, ftraunge Engines and In- ftriimentes ; : for fundry purpofes in the Common Wealth ? or for°priuate plea- hire .? and for the better maintay ning of their owne ehate ? I will not ( therefore) A.iiij. fight lolmDee his Mathematical! Preface. fightagainft myne owne fhadowe. For, no man (lam fure) will open his mouth againft this Enterprife.No ma (I fay) who either hath Charitie toward his brother ( and would be glad of his furtherance in vertuous knowledge) : or that hath any care & zeale for the bettering of the Comon ftate of this Realme.N either any, that niake accompt, what the wifer fort of men ( Sage and Stayed ) do thinkc of them; T o none ( therefore ) will I make any Apologie, for a vertuous a&e doing : and for * c6mending,or fetting forth, Profitable Artes to English men, in the Englifh toung. „ But, vnto God our Creator , let vs all be thankefull : for tha i,jfs he ,of his Good* » neSjby his Town , and in his la if e dome , hath Created all thynges , in Number, S33 35 JV aight^and Meafure\ So, to vs , of hy s great Mercy , he hath reuealed Meanes, }> whereby, to attcyne the fufficient and neceflary knowledge of the forefayd hys ” three principalllnfiramentes : Which Meanes , I haue abundantly proued vnto 33 you,to be the Sciences and i^Artes fJMathematicall. And though I haue ben pinched with firaightnes of tyme:that,no way, I could fo pen downe the matter(in my Mynde) as I determined : hopyng of conucnient layfure •• Y et,if vertuous zeale, and honeft Intent prouoke and biyng you to the readyngand examinyng ofthis Compendious treatife,I do notdoute, but,as the veritie therof(accordyng to our purpofe ) will be euident vnto you : So the pith and force therof , will perfuade you : and the wonderfull if ute therof, highly plea- fure you. And that you may the eafier perceiue,and better remember , the prin- The Ground, cipall pointes , whereof my Preface treateth , I will giue you the Groundplatc jlatt of this of my whole difcourfe,in a Table annexed: from the foft to the lafl,foinewhat Me- pntfaceina thodically contriued. T able. If Haft, hath caufed my poore pen, any where , to Rumble : You will, (I am fure) in part of recoin pence, (for my carneft and lincere good will to plea- fure you) , Coniider the rockifh huge mountaines, and the perilous vnbeaten wayes, which ( both night and day , for the while ) it hath toyled and labored through,to bryng you this good N ewes, and Comfortable profe, of Vertues frute. So, I Commityou vnto Gods Mercyfuil diredion , for the reft : hartety befechy ng hym, to profper your Studyes,and honeft Intentes: to his Glory, & the Commodity of our Countrey. Amen. Written at my poore Houfe At Cfytortlake. Anno. i s 7 o. February.# . f.vee. J « IJ C Ga Here liauc you(accorcling to my promiffe) the Groundplat of’ my MAT HEM ATI CALL- Pradface: annexed to Euclide (now firft) publifhed in our Englilhetounge. An, i 570. Febr. 3. I f _ Sciences, and Artes Mathe- maticalL Erincipdl, Iffbich are two, oneljy Arithmetike.< Simple, tenances ; where, an Fmt, is Indiuijtble . ^Geometric. - ddiX t , Which With aide of Geometric principal, demon Sirateth fome tArithmeticall Con- clufion , or Turpofe, Simple 3Which dealeth With Magnitudes, onely : and demonSlrateth all their properties, pajfi* ens, and appertenances : whofe Torn , is Indmfible . . MlXt, Which With aide of ^Arithmetike principal! , demon ftrateth feme G eometricall purpefi: as EVCLIDES ELEMENTbS. \ " In thinges Supernatu - ' r all, at cm all, & Diuine: By Application, Afcem ding. The vfe thereof, is ^ * either } In thinges Mathema¬ tical l: Without farther Application. J . In thinges If at ur all: both Subflatiall,& Ac- cidentall,Fifible, & In- uifible.&c.By Applicat „tion: Defcending. The like Vfes and zfifippli -= cations- are, ( though in a degree lower ) in the Artes Mathema¬ tical Deri- uatiue. are ^either< ft 1 K \ ' i The names of the Principalis: as,< Arithmetike, f Arithmetike ofmoftvfuall whole Numbers: And of Fra&ions to them appertaining. Vulgar : lohich Arithmetike of Proportions. conJideretlA Arithmetike Circular. | Arithmetike ofRadicallNubers:Simple,Compoimd,Mixt r And of their Fractions. ^Arithmetike of Cosfike Nubers : with their Fractions : And the great Arte of Algiebar. "AllLengthes. Geometric, Twlgarispbich tea. cbetb Measuring' Athand—r < All Plaines: As, Land, Borde, GlaiTe,&c.- w All Solids : As, Timber, Stone,V elTels,&c.“ from the ib ace ng fDeriuatiue fro tbe Princi* pads: of Sohichy Jome bane < How farre7from the CMeafurer , anf thing is: of him fene,on Land or Water: called Apomecometrie, - f f 1 Hovp high or deepe, from the leuell j of the 'JMeafurers Handing , any thing is: j Scene of hym , on Land or Water : called Hypfometrie. Of which are o-rowen 0 the Feates >&• Artes of < r Ho\ V broad 1 a thing is , which is in the Meafurers vcw :foithe ftuated on Land or Water ; called Platometrie. ^ J Mecometrie. <J Embadometrie, Stereometric. Geodefie : more cunningly ts Meafure and Suruey Landes a Woods, Waters.&c. Geographic. Hydrographic. Stratarithmetrie. ' Which demonfrateth the matters and properties of all Radiations SDireBc, 'Broken, and TfefleBed. Aftronomie,— ~ Which demonstrated the Distances, Magnitudes, and all IfaturaH motions, Apparencesjand Tajfons , proper to the Planets and fixed Starres.for any time, paft, prefent, and to come : in refpeBe of a certaine Horizon. or Without refpeBe of any Horizon, Mufike, — — Which demonfrateth by reafon,and teaeheth by fenfe, perfectly to iudge and order the diuerftie ofSoundes , hie or lew. Cofmographie, ■— Which, Wholy and perfectly maketh defeription of the Heautnly,andalfo Elementall part of the World : and of thefe panes, maketh homologall application, andmutuall collation necefary . Perlpectiue,- fPropre names ^ A Aftrologie, Statike, — Whic b reafonably demonfrateth the operations and cjfeBes of the natural! beames of light, and fecrete Inf Hence of the 'Planets ,and fixed Starres , in cuery Element and Elementall body : at alt times, in any Horizon ajftgned. Which demonfrateth the caitfcs ofheauincs and light nes of all t hinges : and of the motions and properties to heauines and lightnes «[[ Imprinted by John Day . An. 1 s 70. Feb.z*. ' : Allthr Op Ogl~ap hie, whic h defribeth the JJnbcr, Meafure, Waight, Figure, Situation, and colour of cuery diners thing contained in the perfetle body of tW zA Ifjandgeueth certaine knowledge of the Figure, Symmetric, Waight,- Char aBerization,& due Locallmotion of any per cell /-p | -t • I of the fay d body afigned: and of numbers to the faid per cell appertaining. 1 rOChUlKe, — — - which demonfrateth the properties of all Circular motions: Simple and Compound. Helxcofophie, — — ■— which demonstrated the defgning of all Spiral! lines : in Plaine,on Gy Under, Cone, Sphtre, Conoid, and Spharoid : and their pro- A perties, Pneumatithmie, — Which demonfrateth by clofe hollow G eometricall figures ( Regular and Irregular ) the firaunge properties ( in motion or Stay ) of the Water, ^Ayre, Smoke, and Fire, in their Continuity ,and as they areioyned to the Element es next them. Menadrie,— — — ~ Which demonfrateth, how, aboue Ifaturcs ZSertue, and power f tuple : ri)crtue and force, may be multiplied : and fo to direBe, to lift, to pull to, and to put or call fro, any multiplied, or fimple determined Vertue, Waight, or Force : naturally, not, ft, dircBible, or I t • t • moueable. Jtlypogeioaie, * Which demonfrateth , how ,vnder the Spharica/l Superficies of the Earth, at any depth, to any perpendicular line ajftgned ( Whofe di¬ fiance from the perpendicular of the entrance : and the Azimuth likew’fe, In refpeBe of the fay d entrance, is knowen ) certaine Way, r T J „ ° mayheprefcribedaridgone,&c. JTiy Q1 agogie,— — Which demonfrateth the pajfible leading of Water by Natures law, and by artificial i helpe,from any head( being Spring, Slanding,or running Water ) to any other place ajftgned. Horometrie, — — Which demonfrateth, hove, at all times appointed, the precife,vfuall denomination 'ftime,m(ty be knowen, for any place ajftgned, Zographie, — ■ Which demonfrateth and teaeheth, how, the Inter feBion of ail vifuall Pyramids, made by any plain e ajfignedC the Centcr,difiancet and light es being determined ) may be, by lines, and proper colours reprefented . Architecture,— Which is a Science garnifed With many doBrines, and diuers InfruBions : by Whofe iudgemcnt,allWorkcs by other Workmen fini- fied, are fudged. Nauip-ation,— — Which demonSlrateth, how, by the Short efl good Way, by the apteft direBion,and in the forteSl time-.afujficient Shippe, betwenea - ^ ny tno places (in pa jjage nauigab le) afigned , may be conduBediand m all formes and naturall disturbances chaunctng , how to vfe — .. the beStpojfblemeanes, to recouer the place firSt ajftgned. i haumaturglke,- which genet h certaine order to make firaunge Workes,ofthefenfe to be perceiuedtand of men greatly to be Wondred at . Archemaftrie, Which teaeheth to bring to aBuall experience fenfble,a!l Worthy conclufions ,by all the Artes Mathematicallpurpofed : and by true Naturall philofephie, concluded: And both addeth to themafarder Scope , in the termes cf the fame Artes: andalfo , by his proper Meehnd. and in peculiar termes,procedeth, with helpe of tbe forfayd Artes ,tothe performance of complete Experiences : Which, of no particular Arte, are hable( Formally )to be challenged. f lohn Dee his Mathematical! Preface . (|Tbc. firft booke of Eu- elides Elementes N this first sooKsis intreated of the moft fimple, eafie, and firft matters and groundes of Geo¬ metry, as, namely, of Lyncs, Angles, Triangles, Pa¬ rallels, Squares, and ParaHelogrammc s . Firft of they r definitions, fhewyng what they are. After that it tea- chcth how to draw Parallel lyncs, and how to forme diuerfiy figures ofthreefides5& foure fides, according to, the varietie of their fides, and Angles : & copareth them all with T riangles ,& alfo together the one with the other . In it alfo is taught how a figure of any forme may be chaunged into a Figure of an other forme. And for that it entreateth of thefe moft com¬ mon and general! thynges , thys booke is more vniuerfall then is the fecondc, third, or any other, and therefore iuftly occupieth the firft place in order : as that without which, the other bookes of JEucUde which follow, and alfo the workes of others which haue written in Geometry, cannot be perceaued nor vnderftan- ded. And forafmuchasallthedemonftrations and proofes of all the propositi¬ ons in this whole booke, depended thefe groundes and principles following,, which by reafon of their playnnes neede no greate declaration, yet to remoue all (be it neuer fa litle) obfeuritie, there are here fet certayne fliorte and manifeft expofitions of them, ' . ^Definitions. i, Afigne or point is that fWhich hath no part The better to vnderftand what matter of thing a figneor point is,yemuft note that the nature and propertieof quantitie(wherof Geometry entreateth )is to be deuided, fo that whatfoeuer may be deuided into fundry partesys called q.uantitie.But a point, a! though it pertayne to quantitie, and hath his beyng in quantitie, yet is it no quanti- tie, for that it cannot be deuided. Becaufe(as the definition faith) it hath no partes in¬ to vv filch it Should be deuided. So that a pointe is the leaf; thing that by minde and vn- derftandingcanbeimagined and conceyLied : then which,th ere can be nothing lefte* as the point A ifi the raargent. The argument of the firfl Book.** T>ef!nit'm) tf A Afign eorpointisof Tithagoras Scholersafterthismanner defined: Apojntism vmtte'tohtch bath pofnistj. Nubers are coneeauedin mynde without any forme & figure, *PV***f*e*’ and therfore wi thout matter wheron to receaue figure, & confequently without place agoras, andpofition. Wherfore vnitie beyngapartofnumber,hath nopofition, or determi¬ nate place. Wherby it is manifeft,that number is more Ample and pure then is magni- tude,and alfo immateriall.* and fo vnity which is the beginning of number, is lefts ma¬ terial! then a figne or poynt, which is the beginnyng of magnitude.For a poynt is ma¬ terial!, and requirethpofition andplace,andtherby differed! from vnitie. 2« A line is length “Without breadth . Def'nitim of * line. There pertaine to quantitie three dimenfions, length,bredth,& thicknes,or depth.* and by thefe thre are all quatities meafured & made known . There are alfo, according B,j, ip I S <tn other defi¬ nition of a line. A n other. The cades of a line. Difference of a point fro Smtj. Vnitie is a fart of number. A poynt is »» pnrt of quart- title. Definition of A right line. Definition of a right line after Campanus. Definitio therof after Archi¬ medes. Defining thertf After Plato. An other def¬ inition. Another. to thefe three dimenfion s, three kyndes of continuall quantities : a lyne, a fuperficies , orplaine,anda body.Thefirftkynde,nameIy,alineis here defined in thefe wordes, <>A lyneis length without breadth. A point, forthatitis no quantitie nor hath any partes into which it may be deuided,but remaineth indiuifible,hath not, nor can haue any of thefe three dimenfions.lt neither hath Iength,breadth,northickenes.But to a line.which is the firft kynde of quantitie.is attributed the firft dimenfion, namely, length, and onely thatjf’orithath neither breadth nor thicknes,but is eonceaued to be drawne in length onely, and by it,it may be deuided into partes as many as ye lift,equall,or vnequall.Bu t as touching breadth it remaineth indiuifible. As the lyne A B, which is onely drawer* iti length, may be deuided in the pointe C equally, or in the point D vnequally,and fo into as many partes as ye lift. There , _ are alfo of diners other geuen other definitions of a lyne: as A c 3 thefe which follow. __ zA lyne is the moiiyng of a poynte,zs the motion or draught of a pinne or a penne to your fence maketh a lyne, Agayne,<*^/ lyne is a magnitude hailing one onely [pace or dimenfion, namely, length Wantyng breadth and thickyes. | T he endes or limites of 4 lyne ^re point es. For a line hath his beginning from a point,and likewife endeth in a point; fo that by this alfo it is nianifeft,that pointes, for their fimplicitie and lacke of compofition, are neither quantitie,nor partes of quantitie,but only the termes and endes of quantitie. As the pointes zAy B, are onely the endes of the line A B , and no partes thereof , And herein differeth a poynte in quantitie, from vnitie in number: for rhat although vnitie be the beginningof nombers, and no _ _ ___ _ ^ numberfasapointis the beginning of quantitie,and no quan- A 3 titiejyet is vnitie a part of number.For number is nothyng els but a colle&ion of vnities,and therfore may be deuided into them, as into his partes. But a point is no part of quantitie,or of a lyne*. nfeither is a lyne compofed ofpointes,as number is of v nities .For things indiuifible being neuer fo many added together, can neucr make a thing diuifible,as an inftant in tiine,is neither tyme,nor part of tyme,but only the beginning and end oftime,and coupleth &ioyneth partes of tyme together. 4 A right lyne is that "Which lieth equally betwene his pointes. As the whole line zA B lyeth ftraight and equally betwene thepoyntes AB without any going vp or comming downe on eyther fide. Campanus and certain others, define a right find thus: A _ A right line is thejhortefl extenfion or draught, that is or may b'e from onepoynt to an other. zArchimedes defilieth it thus. A right line is the fnorteif of all lines, which haue one and the fitlf fame limites or endes: which IS in maner al one with the definitio of Campanus.Ks of all thefe lines A B C7A D C7A E Cs A F C, which are all drawen from the point A7 to the poynte4?£as Campanus fpeaketh, or which haue the - - & felf fame limites or endes,as Archimedes fpeaketh,the t> _ _ lyne AB C, beyng a right line,is the Ihorteft. g tP/Wf<?defineth a right line after this maner: Aright line is that whofe middle part JhadoWeth the extremes. As if f you put any thyrig in the middle of a right lyne,you lhall not fee from the one ende to the other,whxch thynghappeneth not in a crooked lyne. The Ecclipfe of the Sunneffay Aftronomers) then happeneth,when the Sunne,the Moone, &our eye are in one right line.For the Moone then being in the midft betwene vs and the Sunne, caufeth it to be darkened.Diuer s other define a right line diuerfly,as followeth, j tA right lyne is that which fiandeth firme betwene his extremes. Aga ync,A right line is that which With an ether line of lyke forme cannot make a figure. Agayne, of Euclides Elementes . FoL 2. A gay ne,<*^ right lyne it that which hath not one pan in a plain: fuperficies, and an other erected, on high. - Aeayne, Aright lyne is that, all Whofi partes agree together With all his other partes, Agayne,-^ right lyne is that,whofe extremes abiding, cannot be altered. Euclide doth not here define a crooked lyne, for it neded not.lt may eafely be vnder- ftand by the definition of a right lyne, for euery contrary is well manifefted & fet forth by hys contrary. One crooked lyne may be more crooked then an other, and from one poynt to an other may be drawen infinite crooked lynes : but one right lyne cannot be righter then an other, and therfore from one point to an other, there may be drawen but one right lyne. As by the figure aboue fet,you may fee, 5 yfftfper fries is thatftobicb bath onely length and breadth. r A fuperficies is the fecond kinde of quantirie, and to it are attributed two dimenfi- ons, namely length, and breadth. As in the fuperficies <sArBCcD, whofe length is taken by thelyne^A, or CD, and breadth bythe lyne ^4C.or‘2?£Z):andbyreafonofthofetwodimenfions a fuper¬ ficies may be deuided twowayes, namely by his length, and by hys breadth, but not by thicknefie/orit hath none.For,that is attribu¬ ted onely to a body,which is the third kynde of quantitie,and hath all three dimenfions,length,breadth, and thicknes,and may be de- uided according to any of them. Others define a fuperficies thus : A fuperficies is the terme or ende of a body. As a line is the ende and terme of a fuperficies, 6 Extremes of a fuperficies, are lynes. As the endes,limites,or borders of a lyne,are pointes,inclofing the line: fo arc lines. thelimites,borders,andendesinclofingafuperficies. As in the figure aforefayde you maye feethe fuperficies inclofed with fou re lynes. The extremes or limites of a bodye, are fuperficieiles,And therfore a fuperficies is of fomc thus defined; A fuperficies is that , Which endeth or inclofeth a body : as is to be fene in the fides of a die, or of any other body , 7 flame fuperficies is thatftobich lieth equally betwene his lines . As the fuperficies AB CD lyeth equally and fmoothe betwene the two lines AB, and CD: or betwene the two lines AC , and *B 'D : fo that no part therof eyther fwelleth vpward,or is depref- feddownward.Andthisdefmitiomuchagreeth with the defini¬ tion of a right line, A right line lieth equally betwene his points, A B and a plaine fuperficies lyeth equally betwene his lynes. Others define a plaine fuper¬ ficies after this maner: tyi plaine fuperficies, is the fliortesl extenfion or draught from one lyne to an other dike as a right lyne is the (horteft extenfion or draught from one point to an other, Euclide alfo leaueth out here to fpeake of a crooked and hollow fuperfic ies,becaufe it may eafely be vnderftand by the diffinition of a plaine fuperficies, being hys contrary. And euen as from one point to an other may be drawen infinite crooked lines, & but one right line, which is the (hortefi: : fo from one lyne to an other may be drawen infi¬ nite croked fuperficielfes,& but one plain fuperficies, which is the (horteft.Here mu ft you confider when there is in Geometry mention made of pointes,lmes,circles,trian- gles,or ofany other figure$,ye may notconceyue of them as they be in matter, as in woode, inmettall, in paper, or in any fuchlyke, for fo is there no lyne, but hath feme breadth, and maybe deuidedmor points,but that fhal haue fomepartes, and may alfo be deuided,and fo of others,But you muft conceiue them in mynde,plucking them by imagination from all mattcr,fo {hall ye vnderftande them truely and perfectly, in their owne nature as they are defined, As alynetobelong,andnotbroade:andapoynte to B. i j, b« Another An ether, jin ether. Vrhj Euclid* here defneth net a crooked lyne. Definition of A fuperficies. A fuperfic set may he deusded two mujes. An other dofinE tiers of a fsper- ficies. T he extremes of * fuperficies. Another de fin's « tionofn fuper - ficies. Definition of a. plume fuperfic cies , Another definE t ion of n pluynt fuperficies. NOTE , 1 Amthtr defini¬ tion of a flay no fiber fetes , An other defi¬ nition. Another defi¬ nition. An other defi¬ nition. Definition of a playne angle. definition of a r edit lined an- gk* Three fndes of an pies. O VVhat a right angle What alfo a perpendi¬ cular lyne it. TV hat an o&- Sttfe angle it. Si*? The first Tioolg be fo little, that it fhall haue no part at all. Others othcrwyfe define a playne fuperfic ics'.cyf plains Jkperficies is that, which is firmly fit betxvene his extremes, as before wa s fayd of a right lyne. Agayne.e^f plaine Jkperficies is that junto all Whofi partes a right line may Well be applied. Again, A plaine faperficies is that , Which is the fhorteft ofal Jkperficies, which haue one & the felf extremes: Asa right line was the fiiortefi; line that can be drawen betwene two pointes, Againe,Af playne Jkperficies is that, whofi middle darkeneth the extremes, as was alfo fayd of a right lyne. * 1 1 8 A plaine angle is an inclination or bowing of fib o lines the one to the other and the one touching the other ^and not beyng direBly ioyned together. - As the two lines AB,ScB C, incline the one to the o- ther,and touch the one the other in the point®, in which point by rcafon of the inclination of the fayd lines, is made the angle A B C. But if the two lines which touch the one the other,be without allinclinationof the one to the other,artd be drawne dire&ly the one to the other,then make they not any angle at all,as the lines CD, and D Ey touch the one the other in the point D} and yet as ye fee they make no angle. 9 And if the lines 18? hich containe the angle be right lynesjhen is it called 4 rightlyned angle . As the angle A B C,in the former figures,is a rightlined angle, becaufe it is contai¬ ned of right lines : where note,that an angle is for the moft part deferibed by thre let- ters,of which the fecond or middle letter reprefenteth the very angle, and therfore is fet at the angle. By the contrary,a crooked lyned angle,is that which is contained of crooked lines* which may be diuerfiy figured. Alfo a inixt angle is that which is caufed of them both* namely, of a right line and a crooked, which may alfo be diuerfiy figured, as in the fi¬ gures before fet ye*may fee. There are of angles thre kindes, a right angle,an acute an- gle,andan obtufe angle, the definitions of which now follow. io When a right line J landing Upon a right line maketh the angles on either fide e quail: then either of tbofe angles is a right angle. And the right lyne iphkh flandeth ere cle is called a perpendiculer line to that fifion Tehicb it fiandeth. As vpon the right line CD, fuppofe there do fiand an other line A. A A, in fuch fort,thatit maketh the angles on either fide therof e- quall : namely,the angle ABC on the one fide equall to the afigle AB Don the other fide : then is eche of the two angles A B C^and A BID a. right angle, and the line A B, which fiandeth ere&ed vpon the line CD, withoutinclination to either part is a perpendicular line,commonlycalledamongartificersaplumbelyne. c ft 1 1 An obtufe angle is that which is greater then a right angle. As ofEuclides Elementes . FoL^, As the angle CBE in the example is anobtufe angle, for it is greater then the angle A BC, which is a right angle, becaufe it con tayneth it,and containeth moreouer the angle ABE . 12 An acute angle is that &hich isle fie then a right angle. As the angle EB Din the figure before put is an acute angle,for that it is lefle then the angle A B T), which is a right angle, for the right angle contai¬ neth it,arid moreouer the angle ABE. i$ A limite or termejs the ende ofeuery thing . For as much as ofthinges infinite (as TUte faith) there is no fcience, therefore muft magnitude or quantitie(wherof Geometry entreateth) be finite,and haue borders and limites to inclofeit/which are here defined to be the endes therof. As a point is the li¬ mite or terme of a line,becaufe it is thend therof : A line likewife is the limite & terme of a fuperficies : and likewife a fuperficies is the limite and terme of a body,as is before declared. 14 A figure is that which is contayned finder one limite or terme, or many . As the figure A is contained vnder one limit, which is the round line, Alfo the figure 3 is con tayned vnder three right lines. And the figure C vnder foure,and fo of others, which are their li- mites ortermes. 15 A circle is a plaine figure ^conteyned finder one line , "which is called a cir <* cumfierencefmto fiahich all ly^ard%en from one poynt trithin the figure and falling fipon the circumference therof are equal! the one to the other. As the figure here fetis a plaine figure, thatis a figure without groffenes or thick- nes,and is alfo contayned vnder one line,namely,the crooked lyne B CD,which is the circumference therof, it hath moreouer in the middle therof a point, iiiamelyjthgpointe^, from which, all the lynes drawen to the fiip®ISSeS,are equal: as the lines AB,AC> A i>7 and other how many foeuer. Ofall figures a circle is the moft perfect, andtherfore is it here firft defined, 16 And that point is called the centre of the circle , as is the point A, which is Jet in the middes of the former circle . For the more eafy declaration,that all the lines drawen from the centre of the circle to the circumference,are equall, ye muft note, that although a line be not made ofpointes: yet a point,by his motion or draught, de- feribeth a line, Likewife a line drawen,or moued, deferibeth a fu¬ perficies: alfo a fuperficies being moued maketh a folide or bodie. Now the imagine the line A 3, (the point A being fixed)to be mo* ued about in a plaine fuperficies,drawing the point B continually about the point till it returne to the place where it began firft tomoue: fo (hall the point 2?,by this motion, deferibethe circum¬ ference of the circlejand the point tA being fixed, is the centre of the circle. Which in BJii. ali What an aiuft angle is. T he limite of anj thing. No fcience ef t hinges infinite Definition of a figure. Definition of it. circle. A circle the ■moft perfeti ef all figures. The centre ef & circle. Definition of* diameter. Definition of fsmicircl e. Definition of it fction of a cir¬ cle. Definition of recltlmed fi¬ gures. Definition of three fidcd fi¬ gures. 5^ The firU TSoolfe all the time of the motion oftheline,hadlikediftance: from the circumference, name¬ ly ,the length of the line A B. And for that al the lines drawn from the centre tothe cir** cumlerence are defcribed of that line,they are alfo equal vntoit, & betwene thefelues. 17 A diameter of a circle fis a right line^hich dralben hy the centre thereof, and ending at the circumference on either fide, deuideth the circle into tTVo equall partes , As the line B A Cm this circle prefent is the diameter,becaufe it paffeth from the point 3, of the one fide of the circumferece, to the point C,on the other fide of the circumference, & paffeth alfo by the pointed being the centre of the circle. Andmoreo- 3 uer it deuideth the circle into two equall partes.- the one,name~ 1 y B © C,being on the one fide of the line, & the other namely, BE C, on the other fide,which thing did Thales Miletius (which brought Geometry out of Egiptinto GreceJ firft obferue and proue, For if a line drawen by thecentre,do not deuide a circle into two equal partes: all the lines drawen from the centre to the circumference Ihould not be equall* T> IS A femicircle fis a figure Tvhich is contaynedvnder the diameter y andvni der that part of the circumference which is cut of by the diametret As in the circle ABCTt the figure B AC is a fcmicircle,becaufe it is contained of the right line B CC, which is the diametre,and of the crooked line B A C, being that part of the circumference, which is cut of by the diametret G C. So likewife the other part of the circle,namely B D C, is a fcmicircle as the other was. 19 A fe Elton or portion of a circle , is a figure Tvhiche is contayned Vnder a rightly tie, and a parte of the circumference 0 greater or leffe then the fimicircle. As the figure B C, in the example, is a fedion of a circle, & is greater then halfe a circlc,and the figure A 2) C,is alfo a fedi- on of a circle,and is leffe then a femicircle. A fedion, portion, or part ofa circle is all one, and fignifieth fuch a part which is ei¬ ther more or leffe, thenafemicircle: fo that a femicircle is not ^ here called a fedion or portion of a circle, A right lyne drawen from one fide of the circumference of a circle to the other, not pfdlyngby the centre, deuideth the circle into two vnequall partes, which are two fedions,of which that which contayneththe centre is called the greater fedion,and the other is called the leffe fedion. Asinthecxamplc,thepart of the circle^/ B C,which containeth in it the centre E, is the greater fedion,beinggrea ter then the halfe circle: the other part, namely <tA D C, which hath not the centre iia it,is the leffe fedion of the circle,bcing leffe then a femicircle. 20 (bright lined figures are fuch tbhich are contaynedvnder right lynes. As are fuch as followeth,of which fome are contayned vndcr three right lines, fom$ vnder foure,fome vnder fiue,and fome vnder mo, 2 1 Tbre ftded figures ,or figures of threfy desire fuch Tvbich are contay* ned Vnder three right lines , As ofSuclides Elementes . As the figure in the example A B C, is a figure of three fides, becaufeitis cotamed vnder thre right lines, namely,vnder the lines lABfB C,CzA. A figure of three fides, or a triangle, is the firft figure in order of all right lined figures , and therfore of all others it is firft defined. For ynderleffe then three lines, can no figure be comprehended. 22» Foure fided figures or figures offoure fides are finch , lehich are contained ynder foure right lines. As the figure here fet,is a figure offoure fides,for that it is c5~ . ^ prehqnded vnder foure right lines,namely,A B,B D,D C,C A. — — * - — - Triangles,and foure fided figures ferue commonly to manyv- fes in demonftrations of Geometry . Wherfore the nature and L- - - - — - properties of them, are much to be obferued, the vfe of other ft- *“ gures is more obfeure. Foly. A 23* Many fided figures are finch "Which haue mo fides then foure. Right lined figures hauing mo fides then fo\ver,by continual adding of fides may be infinite. Wherfore to define them all feueralIy,accordingto the number of their fides, fliould be very tedious,or rather impoifible. Therfore hath Euchde comprehended the vnder one name,and vnder one diffmition : calling them many fided figures, as many as hauc mo fides then foure : as if they hauefiuc fides,fixe, feuen, or mo. Here noteye9 thateuery rightlined figure hath as many angles,asit hath fides,&takcth his denomi¬ nation afwell of the number of his angles, as of the number of his fides- As a figure co- rained vnder three right lines,of the number of his three fides, is called a thre fided fi¬ gure : euen fo of the number of his three angles, it is called a triangle. Likewife a figure contained Vnder foure right lines,by reafon of the number of his fides, is called a foure fided figure : and by reafon of the number of his angles, ki$ called aquadrangled fi* gure,andfo ofothers. 24. Of three fided figures or triangles } an equilatre triangle is that} -tohich hath three equall fides. • » » . : , • . Triangles haue their differencespartlyof their fides, and partlyoftheirangles . As touching the differences of their fides, there are three kindes.For either all thre fides of the tri¬ angle aj-e eqqalbor two onely.are equall, & the third vnequal : ©r eis all three are vUea uali the one to the other.The firft kind of triangles,namely,that whichhath three equall fidesfis molt fitnple,andeafieft to be knowemandis here firft defined, and • • • • Ban;*. Definition of foure fided figure! . Definition if m.tnj fided figures* Definition of a ntefUtUtSF Hrianglt, Definition of «m» t fio fee let. Definition of ■i* ScnUnttm. An Qrtbigoni- vin t<ri,tnfie. An Oxlgont* Ifnttrfangie* ThefirH'Booke • is called an equilater triangle, as the triangle *A in the example, all whofe fides are of one length. . , ■" ) • ■ . . 'W-'V-';,' " f* 25 . Ifofceksjs a triangle ftohich hath onelj two fides equaU. The fecond kinde of triangles 4 hath two fides of one length, but the third fide is either longer or ihorter the the other two, as are the triangles here figured ,B,C,D In the triangle F,the two fides A E and E F are equal the one to the other,and the fide A F, is 16- ger then any of them both:Likewifein the triangle Cthe two fides G quail, and the fide G K is greater. Alfo in the triangle D,the fides L MnndAlN3aie «. quail, and the fide L Nis Ihorter. 26. S calenum is a triangle fWbofie three fides are alh>nequaU. As are the triangles E,F, in which there is no one fide equall to any of theother.For in the tri¬ angle F,the fide AC is greater then the fide 2? C, and the fide B Cis greater the thefide AB. Like- wife in the triangle F,the fide D H,is greater the the fide B (?,and the fide D G}is greater then the fide G H, warn* An Amtligoni- am triangle. 2 7. Againe of triangles ,an Orthigonium or a rightangled triangle s l angle ’Which hath a right angle. As there are three kindes of triangles, by reafon of the diuerfitie of the fides Jo are there alfo three kindes of triangles,by reafon of •the varietie of the angles. For euery triangle either containeth one right angle, & two acute angles : or one obtufe angle,& two acutei orthree acute angles : foritisimpoffiblethatone triangle fhould containe two obtufe angles,or two right angles, or one obtufe an. gle,and the other a right angle. All which kindes arc here defined. Fir ft a rightangled triangle whichehath in it a right angle . As the triangle B CD,of which the angle B CDfis a right angle. 28. An ambligonium or an obtufe angled triangle $ is a triangle "Which hath an obtufe angle. A sis the triangle B, whole angle AC I>, is an obtufe angle, and is alfo aScale- non.hauing his three fides vnequall : the triangle £, is likewife an Ambligonion* whofe angle EG H, is an obtufe angle, & is an Ifofceles, hailing two ofhis fides equalbnamelyF (JandG H. 2i). An oxigonium or an acuteangled triangle 3 is a triangle "Which hath aU his three angles acute . <6 _ » /te mentes. Foil. juarc* As the triangles A, 71, CM the example, al whofe angles are acute.-of which A is an e- quilater triangle, 2?, an Ifofccles, and C a Scalenon.An cquilater triangle is moftfim pie, and hath one vniforme conftrudrion, and therfore all the angles ofit arecquall, and neuer hath in it either a right angle, or an obtufe: but the angles of an Ifofceles or a Scalenon , may di- uerlly vary. It is alfotobe noted that in comparifon of any two fides of a triangle, the third is called a bafe. As of the triangle ABCin refpeft of the two lines A B and A C,the line B C,is the bafe : and in refpedt of the two fides A C and C B, the line A B,is the bafe, and likewyfe in refpedtof the two fides CB & B A, tfie line A C,is the bafe. 5 o Of foure fyded figures, a quadrate orfquare is that, Tthofe fydes are e* °fs quail 3and bis angles right* As triangles haue their difference and varietie by reafon of their . fides and angles: fo likewife do figures of foure fides, take their varie¬ tie and difference partly by reafon of their fides, & partly by reafon of their angles,as appeareth by their definitions.Thc four fided figure ABCD is afquare ora quadrate, becaufeitis a right angled figure, al hys anglesare rightangles,and alfo all his four fides are cquall. ' •< 31 A figure on the otic fiyde longer ^or fiquarelike, or as fome call it ,4 long nfo'Aon of* fquare, is that ybic h hath right angles, but hath not equall fydes # This figure agreeth with a fquarc touching his angles, in that either of them hath right angles, and differeth from it onely by reafon of his fides,in that the fides thcrofbe not e- quall,as are the fides of a fquare. As in the example,the an¬ gles of the figure ABCD , are right angles, but the two fides thereof A B , and CD, are longer then the other two fides D. 3 2 Rhombus (or a diamonde) is a figure hauing fioure equall fydes Jbut it is De^itiott ^ not right angled. Diamond figure t This figure agreeth with a fquare, as touching the equallitie of lines, but differeth from itin that it hath not right angles, as hath the fquare .As of this figure,the foure lines AB,75C, CD , D A} be e- quall.but the angles therofare not right angles. For the two angles ABC and A D C, are obtufe angles,greater then right angles, & the other two angles B A D and BCD , are two acute angles leffethen two right angles. And thefie foure angles are yet equall to foure right angles: for,asmuchas the acute angle wanteth of a right angle., fo much the obtufe angle excedeth a right angle. a ffifiombaides '■I V' Definition of# eltamondUke fi¬ gure. 'Trapezia or tub la. Definition of Turullell/na, W fiat Petici- ens-are. Thefirsl^oo!^ 3 | hombaides(or a diamond like) is a figure ftohofe oppofite /ides ate e* quail find lehofe oppofite angles are alfo equally but it bath neither e* quail fides fior right angles. As in the figure A B CT), all the foure fides are not equall,butthe two fides AB and CD} being oppofite the one to the other,aIfo the other two fides A Cand B Thbeingalfo oppofite, are equallthe oneto the o- ther.Likewife the angles are not right angles,but the angles CAB , and CD B, are obtufe angles, and op¬ pofite and equall the one to the other. Likewyfe the angles A B ‘Z>,and A C D, are acute angles,and oppo- fite,and alfo equall the one to the other. 34 dll other figures of fours fides bejides thejeyare called trapezia fir tables „ Such are all figures,in which is obferued no equallitie of fides nor angles: as the figures A andi?,in themarget,which haue nei- f - - f ther equall fides, nor equal angles, but are described at all aduen- \ 4 / ture without obferuation of order, and therefore they are called / irregular figures. 1 5 (parallel or equidifi ant right lines are fuchfitohicb being in one and thefelfe fame ftipcr fries, and pro* duced infinitely on both fides 3 doneuer in any part i concurre. As are the lines A 2?,and C D, in the example. 5^ Teti cions or requejles . 1 From any point to any point, to dralr a right line . After the definitions, which are the firft kind of princip!es,now follow petitions, which are the fecond kynd of principles: which are certain general fentences,fo plain, & fo perfpicuous, that they are perceiued to be true as foone as they are vttered,& no man that hath but common fence,can,nor will deny them. Of which, the firft is that, which is here fet. As from the point ex/, to the point who wil de- ny,but eafily graunt that a right line may be drawn^For two points - 4 howfoeuer they be fet,are imagined to be in one and thefelfe fame A B plaine fuperficies,wherfore from the one to the other there is fome fhorteft draught, whiche is a right line.Likewife any two right lines howfoeuer they be fet, are imagined to beinonefuperficies, and therefore from any one line to any one line,may be drawen a fuperficie s . . : 4 2 To produce a right line finite, fraight forth continually , M • - As to draw in length continually the right line AB , who will not graunt.? For there is no magnitude fo great, but that there a. B c maybe a greter,nor any fo litkgbut that there may be a kfle.And aline of Euclides Elemdntes . FoL 6* a line is a draught from one point to an other, therfore from the point B, which is the ende of the line zA 2?,may be drawn a line to fome other point, as to the point C, and from that to an pther,ana fo infinitely. 5 Vpon any centre and at any diJlance}to defcribe a circle . A playne fuperficies may in compafle be extended infi¬ nitely : as from any pointe to any pointe may be drawen a right line, by reafon wherof it cdmmeth to paflfe that a cir¬ cle may be defcribed vpon any centre and at any fpace or diftance .As vpon the centre A}and vpon the fpace A B,ye may defcribe the circle BC,8c vpon the fame centre,vpon the diftance A (Z),ye may defcribe the circle D £,or vppon the fame centre ^according to the diftaunce A F, ye may defcribe the circle F G, and fo infinitely extendyng your (pace. 4 jill right angles are e quail the one to the other. v E B ■a This pcticion is moft plaine, and oftreth it felfe euen to the fence. For as much as a right angle is caufed of one right lyne falling perpendicularly vppon an other,and no one line can fall more perpendicularly vpo a line then an other: therfore no one right angle can be greater the an othenneither do the length or Ihortenes of the lines alter the grcatnes of the angleJFor in the example, the right angle A B C, though it be made of much lon¬ ger lines then the right angle D E F,whofe lines are much (hotter, yetis that angle no greater then the other. For if ye fet the point E iuft vpon the point B, then (hal the line E D,euenly and iuftly fall vpon the line A .5, and the line E A, (hall alfo fall equally vpon the line B C, and fo dial the angle 2) E F,be equall to the angle A B £7, for that the lines which caufe them,are oflike inclination. It may euidently alfo be fene at the centre of a circle. For if ye draw in a circle two diameters, the one cutting the other in the centre by right angles, ye (hall deuide the circle into fowre equall partes,of svhich eche contayneth one right angle, fo are all the foure right angles about the centre of the circle equall. 5 When a right line falling Vpon t'tpo right lines foth make on one & the Jelfefamefyde , the twoinVoarde angles lefte then nvo right angles y then foal thefe tHoo right lines heyng produced at length concurreon that part, in "frhich are the tibo angles lefe then two right angles . As if the right line A 5,fall vpon two right lines, namely,C D and E F, fo that it make the two inward angles on the one fide,as the angles D HI Sc F I H, lefle then two right angles (as in the example they do) the faid two lines C D, and E F, being drawen forth in legth on that part,wheron the two angles being lefie the two right angles con(ift,(hal at legth concurre and meete together: as in the point -D,as it is eafie to fee. For the partes of the lines towardes 2) F,are more enclined the one to C.if* the VFhnt cammt* fentences tore. ’Difference be- twerse petitions common [en¬ tentes^ Thefirfl Hooke the other,then the partes of thelines towardes CPare. Wherfore the more thefeparts are produced .the more they fhall approch neare and neare.till at length they fhal mete in one point. Contrariwife the fame lines drawn in legth on the other fide,fbr that the angles on that fide,namely, the angle CHB, and the angle El A, are greater then two right angles, fo much as the other two angles are lefle then two right angles, lhall ne¬ wer mete, but the further they are drawen,the further they fhalbe diftant the one from the other. 6 That two right lines include not a fuperficies. If the lines A B and ./IC, being right lines,fhould inclofe afuperficies,theymufteof ncceffitie bee ioyned together at both the endes,andthe fuperficies mull be betwene the.Ioyne them on the one fide together in the pointeA, and imagine the point.# to be drawen to the point C, fo fhall the line ABi fall on the line A C,and couerit,andfo be all one with it, and neuer inclofe a fpace or fuperficies. 5-^ Common fentences . i Thinges equal! to one and the felfe fame thyng: are squall alfo the one to the other* After definitions and petitions.now are fet common fentences,which are the third andlaft kynd ofprinciples.Which are certaine general propofitios, commonly known ofall'men.of themfelues moft manifeft & cleared therfore are called alfo dignities not able to be denied ofany .Peticions alfo are very manifeft,but notfo fully as are the ed¬ ition fentences.and therfore are required or defired to be graunted. Peticions alfo are more peculiar to the arte whereof they are*, as thofe before put are proper to Geome¬ try: but common fentences are general! to all things wherunto they can be applied, Agayne. peticions con fift in adions or doing of fomewhat moft eafy to be done : but common fentences confift in confideration of mynde, butyetoffuch thinges which are moil eafy to be vnderftanded, as is that before let. As if the line A be equall to the line B, And if the line C 3 be alfo equall to the line By then of neceffitie the lines y ' ^andCjllialbeequaltheonetotheother.Soisitinall ^ <- fuperficielfes,angles,&numbers,&inallotherthings * - ; - * 1 — - — r* (of one kynde) that may be compared together. And if ye adde equall thinges to equall thinges: the fbhole Jhalhe equall. ■B E V F As if theline AB be equal to the line C P>,& to the line A By be added the line B p,& to theline C2?,be - - added alfo an other line E> Pfoeing equal to the line c B E,(o that two equal lines, namely, 5 P,and T> P,be * - added to two equall lynes lAB,& C Pbthen foal the w hole ly ne iAEy be equall to the whole lyne CP, and fo of all quantities generally, I And if from equall thinges 3ye take al&ay equall thinges: the thinges re* may mng fl? all be equall ofEuclides Elementes < Eolj. £ 3 c F As if from the two lines AB and CD, being equal, ye take away two equalLlhies,namely,£ B and F D, then maye you conclude by this com¬ mon fentence,that the partes remayning,name~ ly,^ F,and CF are equall the one to the other.’ and fo of all other quantities . 4 .And if from Vnequall t hinges ye take away equall thinges : the thynges which remayne f mil he Vnequall, E 3 F As if the lines A 2?,and CD, be vnequall,the line <iA 2?, beyng ^ longer then the line C D, &ifye take fro them two equall lines, £~ as£F,andFD:thepartesremayning,whichare thelines a/FF *— and C F,lhall be vnequall the one to the oth er,namely, the lyne %A F,lhall be greater then the line C F, which is euer true in all quantities whatfoeuer* 5 And if to Vnequall thinges ye adde equall thinges : the whole fhall he Vn* equali Asifyehauetwo vnequallines,namely,^Fthegreater,and A e B C F the lelfe, &ifye adde vnto the two equall lines, EB &FD, ~~ thenmaye ye conclude that the whole lines compofed are vn- £ - — ■£* equall : namely, that the whole lyne <sA 2?, is greater then the whole line C D,and fo of all other quantities. 6 Thinges which are double to one and the felfe fame thing: are equall the one to the other. A. . t — - 3 As if the line &A B be double to the line E F,and if alfo the lineCD,be double to the fame line EF: themayyou bythis common fentence conclude, that the two lines vA i?,& C D, £ _ F areequalltheoneto theother.Andthisistrueinall quanti- c ^ ties,and that not only, when they are double, but alfo if they - - - - - be triple or quadruple, or in what proportion foeuer it be of the greater inequallitie. Which is when the greater quantitie is compared to thelefle. 7 Thinges which are the halfe of one and the felfe fame thing: are equal the one to the other. A F As if the line A B ,be the halfe of the line E F, and if the lyne C D, be the halfe alfo of the fame line EF: then may ye con¬ clude by this common fentence,that the two lines c AB and CD, are equall the one to the other. This is alfo true in all kyndes of quantitie,and that not onely when ir is a halfe, but alfo if it be a third ,a q uarter, or in w hat proportion foeuer it be of the lelfe in equallitie. Which is when the lelfe quantitie is copared to the greater* p v, 8 ThinoesWi C3 jjjm together: We equall the one to the other . Such thinges are fayd to agree together,whiche when they are applied the one to the other; or fet the one vpon thfeother,the one excedeth not the other in any thyng, C.iij, As What proporti¬ on of the grea¬ ter inetpttalitj k What proporti¬ on of the lejfe tnequahue is , Whitt a L'ropo- Jirien is. Propofitions of two fortes. What ^ Pro- hit tacit. WhaSa Thee- remeis. TheftrSlBooke As if the two triangles ABC, and DE F, were applied theone to the other, and the triangle AB C , were fet v- pon the triangle 2) £ F,if then the angle A , do iuftly a- greewith the angle F>,and the angle S>with the angle E, and alfo the angle C, with the angle F : and moreouer if the line A F,d o iuftly fall vpon the line T> £,and the line A C,vpon the line D F, and alfo the line 2 C, vppon the line E E, fb that on euery part of thefe two triangles, there is iuft agreement, then may ye conclude that the two trianglesare equalL 9 Euery Thlxple is greater then his part. As the whole is equal to all his partes taken together, fo is it grea¬ ter then any one part therof.Asif the line CB be a part of the line A 4 '5, then by this common fentenceye may conclude that the whole ~ line A B, is greater then the part, namcly,the the line CF,And this is gencrall in all thinges. He principles thus placed tended, now follow the propofitions, which are fentences fetforth to be proued by reafoningand demonftrations, a nd thetfore they arc agayne repeated in the end of the dem onftration. For the propofition is euer the conclufton, and that which ought to be proued. , , , , Propofitions are of two fortes, the one is called a Probleme, the other a Theoreme.1 AProbleme,is a propofition which requireth fome a<ftion,or doing: as the makyng of fome figure, or to deuide a figure or line,to apply figure to figure,to adde figures to- gether,or to fubtrah one from an other ,to deferibegto inferib^ to circumfcribe one fi¬ gure within or without another, and fuche like . As of the firft propofition of the firft booke is a probleme, which is thus: Vpon aright line geuennot bang infinite, to defertbe an e- <■ quilater triangle, or a triangle of three equallfides . For in it, befides the demonftration and contemplation of the mynde, is required fomewhat to be done: namely, to make an equilater triangle vpon a line geuen. And in the ende of euery probleme, after the de¬ monftration, is concluded after this manner, Which is the thing, Which Veas required tobe done* A Theoreme, is a propofition,which requireth the fearching out and demonftration of fome propertieor pafiion of fome figure: Wherinis onely fpeculation and contem¬ plation of minde, without doing or working of any thing. As the fifth propofition of the firft booke,which is thu s,zAn lfofceles or triangle of typo e quail fides,hath his angles at the hafe,e quail the one to the other,&c.isa. Theoreme. For in it is req uired only to be pro¬ ued and made plaine by reafon and demonftratio, that thefe two angles be equai!,without further working or doing. And in the ende of euery Theoreme, after the demonftration is concluded after this ma¬ ne r. Which thyng Was required to be demonstrated or proued . ofEuclides Elementes . Fol. 8 . 54^ The fir ft Trobleme . Ehe fir ft Trnpofition » * Upon a right line geuen notheynginfimtey to deficrihe an e* qnilater triangle yr a triangle of three equall fide s. qnilater triangle ynamely, a triangle of three equal l fdes.'N.o'to the r fore making the centre the point A \and the /pace A Bfefcribe(by the third petition ) a circle B C "D'and agayne (by the fame) tnakyng the centre the point B yand the f pace Bj{yde/cribe an other circle JICB. Andfby tbefirfl petition ) from the point C,wherin the circles cut the one the - ,, other firaTb one right line to the point A,and an other right line to the point B.And for af much e as the point J. is the centre of the circle CBt>^ therforefhy the if. definition) the line A Cise* quail to the line A B : Agayne forafmuch as the point B is the centre of the circle CAB , ther* fore (by the fame de finition) the line B C is e* quail to the line B A. And it is proued , that the line AC is e quail to the line A B : Tbherfore ei * tber of the/e lines C A and CBfis equall to the line flBxbut tbingeslbhich are equall to one and the fame things realfo equall the one to the other (by thefirjl common fentence)l»her fore the line C Ay alfo is equall to the line C B , VV her fore thefe three right lines C AyA By and B C areequalthe one to the other.VVher fore the triagleA B C is e qnilater yV her » fore Vppon the line ABy is defcribed an equilater triangle jdBC. Wherfore Vppon a line geuen not heingin finite y there is defcribed an equilater triangle % Which is the thing ftvbicb 'toas required to be done. A triangle or any other re&ilined figure is then faid to befet or defcribed vpon aline,when the line is one of the fides of the figure. This firfl: propofition is a Pro£/<?;»e,becaufe it requireth atte or doy ng, name; ly,todefcribe a triangle. And this is tobenoted>thateuery Prop fit ion, whether it be a Probleme,oT a T hear eme, commonly containeth in it athi»vgenen ,and athingreqni* red to befiarchedont : although it be not alwayes fo. And rhe thing geuen, is euer fet be¬ fore ihething reejuiredAn fome propofitions there are more things geuethen one, and mo thinges required then onejnfome there is nothing geuen at all, Movcoiier euevy ‘Problems &Theowtte, fay ng perfect and abfolute, ought to haue all thefe partesanamely , F irh the Tropfimn, to be proued. Then the expfimn ^hich Cmflruclien. Demonttratk® Thing geuen. T hwg required Tropfititm . Exgoftten. Dacrmin.ttios). 'Ccnfirucrion. Demwfiratiort. Cendufion. Cafe. The thing geuen ■in this Pro- ileme. The thing required. The proportion. The exposition. The de terms - nation. The conjiructso The i temonfira - turn. T he particular cenclufion. The'vniuerfatl cenclufion. The note where hj it is kgmvne So he a Pro- ilcme. Mo cafes in thjs grope fit sen. Three k fades of demottf ration. ThefirHTiooke which is the explication ofthe thing geuen. After that followeth the determination which is the declaration ofthe thing required.Thenis fee the conffruclion of fuche things which are neccffary ether forthc doingofthc propofitio, or for the demo ftration*Afterwardfolloweththe^w<?»/?r4f/w, which is the reafon and proofe of the proportion* And la.lt ofall is put the concluficn,^ hich is inferred 8c proued by the demonflration,and is euer the propofition,Butull thofe partes are not of ne¬ ed fit ie required in cuery Problems and Theoreme,But the Propofition, demon flra- t ion, and conclufon, are necrilary partes ,8c can neuer be abfent: the other partes may fometymesbeaway. Further in diuerspropofitionsathere happen dtuers cafes: which arenothing els, but varictie of delineation and conftruftioiijor chaunge ofpofition, as when pointes, lines. fuperlicieffes.or bodies are chaunged. Which thingeshappen in diuers proportions. _ ' O w then in this Probleme,ihcthinggeuen,\sthe line gene: the thing required, to be lerched out is,how vpo that line todefenbean equilater triangle. The Pro- pofitiafi of this 'Problems is fUpon a right line geuen not b eyng in finite ft o defer ibe an equilater tri¬ angle. T he expofition [sfuppof that the right linegeuen be A B, and this declarech onely the thing geuen ff \iq determination i s fit is required vpon the line oA B, to defer ibe an equilater mangle: for therby as you fee, is declared onely the thingrequired. The conflruttisn be- gi line th at thefe wo tds^TYjnV therfore making the cetret he point A,&the fpacc A B, defiribe (by the third petkion)a circle &c, and continueth vntil y ou come to thefe wordes, forafinuch as the point A e^c.For thetheitoarc deferrbed circles and lines, neccffary e both for the doyng ofthe proportion, and alfo for the demonftration therof, V vhich demonfirationhegumeth at thefe wordes: eAndforafmuche as the point A is the centre ofthe circle CB D &c: And fo proccdcth till you come to thefe wordes, PDher- fore vpon the line A Bis deferibedan equilater triangle A B C. For vntlll y OU come thether is, by groundes before fee and conftruftions had>proued,and made euident, that the triangle made, is equilater. And then in thefe wordes, therfore vpon the line eA% is defribed an equilater triangle cA B C, is put the firft conclufon. For there are common¬ ly ineuery propofmon two conclufions: theoneperticulcr,the othervniuerfal: and from the firft you go to thelaft. And this is the firft and perticuier conclufi- on,for that it concludeth, that vpon the lyne AB is deferibedan equilater tria- gle,which is according to the expofition*After it,followeth thelaft and vniuerfal conclufon, therfore vpon a right linegeuen not being infinite is deferibedan equilater triangle. For whether the linegeuen be greater or leflethen thys lyne, the fame conftru&ios and demonftrarions proue thefame conclufion. Laft of ail is added this claufe. Which is the thing which 'fyas required to be done: wherby as we haue before noted, is de¬ clared, that this proportion is zProblemeandnot afWw.As for varictie of ca¬ fes in this propofition there is none, for that the line geuen, can haue no diuerfi- ticof pofition. As you haue in this Probleme fene plainelyefetfoorthe the thing geuen, and the thing required, rn O r e o u e r the propofition, expofition, determination, conslrullion, demon firation, and conchifiorfi i eh are general! alfo to many other both Problemes and T heorernes ) fo may you by the example therof diftinft cheai,andfearche them out in other Problemes ,and alfo T heorernes. This alfo is to be noted,that there are three kyndcs ofdemonftrasion. The one is called Demonflmio a priori ,0V compofmon. The other is called Dmomhmw s g po ft mcri* ofEuclitles Elementes. FoLp, .J #pofieriori,oT refoltition. And the third is ademonftracion leadyng to animpof- fibilirie. A demonftration a priori, or cofopofition is, 'when in reafoning, from the prin- Detmnftratu* dples and firft groundeSjWepaffedifcending continually ,till after many reafons c°m' made, we come at the length to conclude that, which we firft chiefly entend. And this kinde of demonftration vfeth Euclide in his bookcs for the moft parr* A demonftration refolution is,when contrariwife in reafoning, we paffe from the laft conclufion madeby theprcmiffes,andby thepremtffes of the premiffesprontinually afcending,til we come to the firft principles and grounds, which, arc indemonftrable,and for they r fimplictty can buffer no farther refolu- tion* '■ -:<c' : " D emcnWratiorx a pc.jlertori }or reflation. A dcmonftration leadyng to an impoffibilitie is thatargumenqwhofe c5- Dcmmfiratio ? clufion is impoffible: that is, when itconcludeth directly againftany principle, oragainft any propofition before proued by principles, orpropofmons before ‘ //-7' ■ proued, Premiffes in an argument,are proportions goy ng before the conclufion premifes what by which the conclufion is proued, th*j«re. Compofitionpaffethfrom the caufeto the effe£t,or from thingcsfimple to thinges more compounded. Rcfolution contrariwifepaffeth from thinges com¬ pounded to thinges more fimple,or from the effect to the caufe. Gompofitionorthefirftkyndeofdemonftration, whichpaffeth from the, principles,may eafely be fene in this firft propofition of Euclide* The demon- ftration wherofbeginneth thus* Andforafmuch a-s the point A is the centre of the circle C B D,therforetheline AC, is equal to the line AB.This reaibn(yoa fec)takcth hisbegtnnyng ofa principle, namely ,ofthe definition of acirde.And this is the firft reafon. Agayneforafmuchas B is the centreof the circle C A E, therfore the line B C is equalltothelyne B A:which is thefecond reafon. And it was before prouedthat thelyne A C is equal! to the line A B, wherforeeither ofthefe lines C A Sc C B is equal to the lyne.A.3 .And this is the third reafS* .‘But things which are equall to one & the felfe fame thy ng, are alfo equall the one to the other. Wherfore the line CAis equal.to the line CB. a fid this 'is the fourth argument. VVherforethefe three lines C A, A B, and B Care equall the one to the other which is the conclufion, and the thing to be proued. You may alfo in the fame; firft Propofido,eafely take an exaple of Refokirid: vfing a contrary order pafify ng backward fro the laft conclufio of the former de- monftration,til you come to the firft principle or ground wheron it began,- For the laft argument or reafon in compofitioh.,is the firft in Refolution: Zc the fir.fi: in compofitlohjis the laft lnxefolution.T hu$ iherfore muft ye precede* The tq flAgle .A B,C is contained ofthree equall ri ght lines, namely j A B,A C,andfi Cj and therfore it is an equilater triangle by the definition of an equilater triangle: and this is the firft reafon. That the three lines be equall, is thus proued. The line s A G and C B are equall to the line A B, wherfore they are equall the one to the other: and this is thefecond reafon. That thelines AB andB C, are equal is thus proued: 1 he lines A B a;id A'C* are draw'en from the centre of the circle A C E,to the circumference .oTpReTame: wherforethey are equall by the dcfinitio ofa circle: and this is the third reafon* Ukewife char thelines A C and A B, are » - ■ ■ D * i, equall An example of conrpeftton m the firji propoft tsen, Tirjl reafon. Second reafn. Third reafon* fourth reafon. Conclufion. Example of re* folution tn the frrft prcpepciofti ftrft reafon. Second reafo n, T bird reafon, fourth reafon which ts the end of the tv hole rent f inti on. •ft ore to deferibe txn ifcjcelei £rj/i,<;g:c. Hftv to describe •«t Sc Men urn. " V~hefirftcBoofy equall,is proued by the fame reafon. For the lines A C and A B arc draVn froth the centre of the circle B CD: wherforcthcy areequallby the fame definition oia circle: this is the fourth reafon or fillogifme* And thus is ended the -whole refolution : for that yon are come to a principle.which is indemoftrablc. 8£ can notberefolued. Ora iemonfirationleadingtoan impoffibiIitie,orto anabfurditie,you may hauean example m the fourth proposition ofthis booke*. RVt nowc if vpoft the fame line geuen, namely, tAB^yc wil deferibe the other two kinds of triangles, namely,an ififceles or atriagleoftwo equal fides,&aScv2/f»c’w,or a triangle of three vnequall hdes.Firfl for the deferibing of an Ifofctln triangle produce the line AB on ether fide,vntill ic concur with the circumferences of both the circles in the pomtes D an d i^/and making the centre the point .^deferibe a circle H F t/ ac¬ cording to the quatity pf the line A F. Likewile making the centre the poynte i?,defcribe acircle HCD G, according to the qua n tide of theline B D . Now the thefe circles ihaii cut the one the other in two poyntes , which let be Hy and 6“: And let the endes of the line geuen be ioyned with one of the fayd factions by two right linesjwhich let be *X G and B G. And forafmuc'heas thefe two lines AS and A D are drawen fro the centre of the circle CDE vntothe circumfe- rece therof,tberfore ar they eq ual . Like wife. the lines B A and B A, for that they are drawen from thecentre of thccir- cle E <lA CF to the circumference ther- , of, are equal, And forafmuch as ether of the lines aA Dznd B Fis cquall to the line ^i?,therfore they are equal theohe to the otheivWherfore putting the line A B comoto the both, the w hole line B 2) fhal- b’e equal! to the whole line *A F.V>i\tB Dh equal to B G,for they are both drawen fro the cetre of the circle HD G to the circumferece therof.Andlikewife by the fame rea- fen the line AFis equal to the line tA' G. Wherfore by the como fcntece the lines A G and B Cj are equal the one to the other, and either of them is greater then the line A B, for that either of the two lines B Z> and Fis greater then the line AB, Wherfore v?, pon the line geuen is defcri’bedan ijhfteks or triangle of two equall (ides. Ye may alfo deferibe vpo n the felfe fame line a Scaleuon , or triangle of three vne- quali fides,if by two right lines, ye i:oyne both the endes of the line geuen to fome one point rhatis in the circumference of one of the two greater circles ifo that that poynt benot in one of the two fections,and that the line 2) F do not concur with it, when it is on either fide produced contlnuallye and dire&lyc.For let the poynte if.be taken in the circumfcreace of the circle HD G} and let it. not be in any of the lections , neyther let the line I? A" concur with it, when it is produced continually and dire&ly vntothe circumfe/encetherof. And draw thefe lines A if and i? if, and the line zA K jhal cut the circumference of the cixdzHFG . Let it cut it in the poynte L : mow then by the common fentence the line B K fhalbe equal to the line aA L,for(by the definition ofa cirdekhelinei? if is equall to theline B 6',and the line A L is equall to the line A d which is equal to the linei? G.Wherfore the line <lA if is greater then the line B K an 3 by the lame reafon maye it be proned that the line B if is greater then the line a AB. Wherfore FoLio. Wherfore the triangle AB K cpnfifteth of three vnequal fides. And fo haue ye vp6n the line geuen,defcribed all the kiudes of triangles. • • This fotobe noted, that if a man will mechanically and re,. ^ defy, not regarding demonftratson vpon a line geuen defcf'xbe a triangle of three eqaall fides, he needethhot to deferibe the whole forelay d circle, but onely alirtle part of eche: namely, wherethey cut the one the other, and fo from the point of the fedion to draw the lines to theendesoftheiinp g.euen. as in this figure here pur. And likewife,if vpon the fa id line he will d'efcribea trian¬ gle of two equal l fydes, let him extende thecompaffe accor¬ ding to the qnantitie that he will haue the fyde to be, whether longer then the line geuen orfhortcr*. andlo.dtaw onely a ii- tie part ofcchecirclc,where they cut the one the other, be fro the point of the fedion draw the lines to the eude of the line geuen. Asia the figures hereput. Note that in this the two fydes muftbc fuch3thatbeyngioynedtogether3 they be lon¬ ger then the line geuen* And fo alfo if vpon the fay d right line he will deferibe a triangle of three vnequal fydes ,let him extend the compare. Firft, according to thequantitiethathe will haue one ofrbe vnequall fydes to be, andfo draw alittle part of the circle,sc then extend it according to the quantum that he wi! haue the other vnequal fyde to be, and draw likewyfe'a little part of the circle, and that done, from the point ofthefedion draw the A 3 lines to the endes oftfreline geuen,as in the figure here put.Notethat in this the7 two fides mud be Inch, that the circles deferibed according to their quatitie may- cut the one the other. i^FMficpnd TroMeme. cFhefecond Tmpofition* Fro a point geuen Jo draw a right Ime equal to a rightline geuen* ofe tbat the point geue be A let the right linege • uenhe'B CJt g required fro [the point A}to \drdwe a right lyne e quail to _ __ J he line BC\ * (Draft (by the fir ft pettcio) from the point A to the poynte B a right line A B: and Vpon the line A B defer ibe(by the fir ft propoftt'to ) an equilater tri * angle ^and let the fame be D A B^and extedjby thefecond peticio ythe right lines D A & DB^ to the poyntes E •£>•*/. and i] Hew to definite an equilater tri angle redilj met bank allj* How to deferibe an Ifofceles tri* angle redilj , Hew to deferibe a Scalenum trs-* angle rtdtlj. Ctnflruttktt, ■ TPgmenjtratiom Two thiriges ge- 'sten in this pro- tfcsin this prspoftion. portion rower c The fir ft cafe. The fecond cafe. The third cafe. TbefirB cBoo%e mdt\(s'(hy the third petkio)making thecentre (Band the/pace B Cdefiribea circle CG H:& agatne( by the fame /making the centre !D and the fpace 3) G defcribea circle G h\Lt And fora/* much as the pointe B is the centre of the circle C G H,therfore(by the i>, definitio) the line BC is equal to the line B G'.and forafmuch as the poynt 3) is the centre of the circle G KjL: therefore (by the fame)the line 3) L is equall to the line 3) G:of 'tohicbtke line 3) A is equall to a line 3) B( by the propofitio going beforefi'rherfore the rejiducyna m elyfhe line A L is e* qual to the refidue, namely }to the line B G(by the third common fentence) And itisproucd that the line B C is e* quail to the line B GVlf her fore eyther of thefe lines ALtr BCis equal to the line B GjBut things which are equall to one and the fame thing are alfo equall the one to the other (by the fir ft commo jentence.)VVherfore the line A L is e* qual to the line B C.fi /her fore from the poynt geue f tamely yA/s drawn a right line A L equall to the right linegeuen B C; which l?as required to be done . f ^ v, . J ' - * . ) , * f # '■ r; . : f; i - , OfProblemes and Theoremes,as we haue before noted,fome haue no cafes at all* which are thofe which haue onely one pofition and conftruction:and other fome haue many and diners cafes:which are fuch propofitions which haue diuers deferiptions 8c coniirudions,and chaunge their pofitions . Of which forte is this fecond propofition, whichis alfo aProbleme.Thispropofitionhath two thin ges geuen :Natnely,a pointe, and a line; the thing required is,that from the pointe geuen wherefoeuer it be put, be d raven a ine equall to the line geuen. Now this poynt geuen may haue diuers pofitios For it may be placed eyther without the right line geuen3or in fome point in it. If it be without it, either itis on the fide of it,fo that the right line drawen from it to the ende of the right line geuen maketh an angletor els it is put directly vnto it, Co that the right line geuen being produced lhallfall vpon the point geuen which is without,Butif it be in the line geuen,theneitherit is in one of the endes orextreames thereof : or in fome place betwene the extremes.So are there foure diuer s pofitions of the poynt in relpedfc of the line. Wher upon follow diuers delineations and conltruftions, and confequent* iy varietie of cafes. For the firfl; cafe the figure before put,feructh. To the fecond cafe the figure hereon the fidefetbelongeth. And as touching the or¬ der both of conftru&ion and of demonftra- tion it is all one with thefirft. The third cafe is eafieft of all,name!y,whe the poynt geuen is in one of the extreames. As for exaple.,ifit were in the point C, which is o/Eudides Elementes * Fol.n. is one of the extreames of the line B C . Then making the centre the poynt C,and the fpace C£ defcribeacirclei? L G\ and from the cen¬ tre C drawe a line vnto the circumference, which let the CZ,,which by the definition of acircle^lhalbe equall to the line geuen B C. The fourth cafe as touching conftru&ion herein differeth from the two firfte , for that whereas in the you are willed to draw a right line from the poynt geuen,namely , A, to the poynt B which is one of the endes of the line geuShereyou fhal not nede to draw that line, for that it is already drawemAs touching the reft,both in conftru&ion and demonftration you may proceede as in the two firfte. As it is manifefte to fee in thys figure here on the fide put. This; propofition for the playnes & cafi- nes thcreofifeemeth to be as it were a princi¬ ple, and may eafly mechanically be done. For opening the compafle to the quantitve of the line geuen , and fetting on foote of it fixed in the poynt geuen and marki ng with the other another poynt wherfoeueritfall, & fo by the firft peticion drawing a right line fro the one of thofe poyntes to the othcr,the fayd righte line fhall beequall to the right line geuen : yet in deede is it no principle, for that it may by demonftration be proued: but principles can not be proued,as we haue before declared. 5^ The 3, Trobleme. T he \fPropofitm. T wo unequal right lines beinggeuenjto cut of from thegrea* ter,a right lyne equall to the lefle. \Vppofe that the tipo Unequal right linesgeuen be AS O' Cy ofiphich let the lyne AS be the greater. It is requi * red from the line AS beinv the greater yto cut of a right line equal to the right line Cyphic-h is the lejle HneAralPf m(hy the Jecond propofition) fro the point A a right line equall to the line Cyand let the fame be A D:and making _ \the centre Aland the fpace AD deferibe (by the third peticion) a circle DEF. And forafmuche as the point A is the centre ofy circle ID E F, therfore A E is equal to A Dfut the line C is equal to the line A DyVherfore either of the fe lines A E andC is equall to A Dy therfore the line A E is equall to the line Cypher fore tlvo Pne quail right lines being geuen gamely, A S and Cohere is cut of from AS being) greater, a right line A E equall to thelejfe b/ ljne3 namely % to C: yphicb Spas required to be done. aih\ This ) The fourth tafe. T his prspeftian though it be rj eafte to be done mechanic tally, jet isn« principle. / Sfjpo tbinges ge- teen tn this pro- paftion. Diners cafes in it. viW v-.vVC The frfl cafe. T he feeond cafe. Th e third cafe „ "TheJirHTSoo^e This pro.pofitionpwhich is a Probleme,hath two thinges getter), namely, rare* vnequali right lines: the thing required isyfrorrithe greater to cut of 'a line equal to the idle. It hath alio diuerscafes.FoTtheliites geuen either are diflin&th’onc from the other: or are ioyned together at orieb'ftheir endes: or they cut. the one' the otber,orthe onecutteth the other in one of the exxreames. V Vhich may be two way es. For ether the greater cutteth the lefTejOr the ieifethe greater.If they cut the one the other, eitherech cuttcth th’other into equali partes ? or into vne- quall partes : or the one into equali parses ,and the other into vnequali partes. V V’hieh may happen i n two forts* for the greater may be cut into equali partes,, and the idle into vnequali partes: or contrariwife* j * ‘ , TO \> 07 v . lor V’ ■ .• Wh en the vnequali lines geuen are diftind the 0ne hom the other , the figure be¬ fore put ferueth. If they be ioyned together at one of their ‘ ends,itis eafieto do. For making the centre that' end where they are ioyned together , & the fpace the le{feline,defcribe a circle: whiche fhall of ne- ceffitiefby the definition of a circle ) cut offiom the greater line a line eq nail to the lefie line , as it ' is playne to fee in the figure here put. . But if theonecut the other in one of the ex-* tremes. As for exaple : Suppofe that the vnequali right lines geuen be A B and CD, of which Jet the . line CD be the greater : And let the line. CD cut the line A B in his extreameC. Then making the centre A and the fpace A i?,defcribe a circle B F. . An4 vpon the line AC describe an equilater tri- , anglefby the firfljwhich let be A E C:8c produce the lines E A and E C, And againe making the ce- tre E and the.fgacevE £ defcribe a circle G £.I,ike« - 1 wife making the centre C and the fpace C<j,def- ' cpibe a circle G' Z-.Now forafmuch .as theJine'A F, , is eq hah to the litre £ G (For Eis the- centre) of f which the line EA is equali to the iine£'C : ther fore the refidue A F is equals to the' refidue C G. Bps. the line ist equali tothq. line ABfov A is the centre, wherefore alfo the line CCj is equal) tothe1me.i4£.Butthe line C G is alfo eqn'all to theline C L^ot the point Cis the centre. Where- . The f earth cafe The fifth cafe* The f set cafe. But now let CD be IefTe then A B,znd let it cut A B, y his extreameC ! Now then cythe-rit cuttein ltiq the - in iddeft or not in the middelt.Frrft let it cut itin the niid deibthen CDis ether the halfe q£A,B& fo is ./fCequal to CD. Or it is leffe then the halfe : and then, mahiqg th? centre C& the fpace C D defcribe a cH!Fe?eiwhich fhTll; cut1 of from the linear/ B a line equ^t.b^j^be'.lifte.C^.nHsi': r . ■ Or it is greater then the half. And the vntp the point J A put the line A F equali to the line CDyby thbiiecotid. ’ ’ And making the centre A & the fpace A F deferibe a cir¬ cle, which ihallcutoffrom the line A B a line equal! to the line -^F.thatisvvnto the line CD, T £u£ , i<n "t 0, Fol.n. But if the line C D do not cut the line AB in the midft: C D fell either be the hake of the line J.B\ or greater then the halfe, or leffe . If C D be the halfe of A B, or leffe then the half of A 2,the making the centre C,and the fpace C D deferibe A a circle whiche fhall cut of from the VmeABa. line equal to the line CD. But if it be greater then the halfe,then againe vnto the point tA put the line A F equal to the line C © (by the fecond propofitio: ) & making the centre A, and the fpace A F defer ibe a ctr chr which {hall cut of from the line AB a line equal! to the line iA F, that is,to the line C D . But if they cut the one the other as the lines C D8zAB do.The making the cetre B & the fpace B A deferibe a circle A F,& draw a line from the point B to the point C,& produce it to thepoint F.And forafmuch as the two right lines B F and CD are vnequalband the lined) cutteththc line BF by one of his extreames, therefore it is ^ poffibleto cut of fromCD aline equall to the line BF. For how to do it we haue before decla- red,wherefore it is pofllble from the line C Dto cutofaline equall totheline AB:o:AB and BF a,te equall the one to the other. ftrufhon and demonirracion,proccde as in the hr it cale. ror it 1? ^ pofition to put to the ende of the greater lyne a line equal! to the leffe lvne , and fo maky ng the centre the fay d eude,and the fpace the leffe linc,to deferibe a cir* cle, which iliall cut of fro tOc greater lyne a. lyne equall to the line put , namely, to the leffe line geuen,as it is manifeft to feei'h the figures partly fiere vnder fee. The feuentk QT eight cafes. The ninth exfi. The tenth e*fe. InnUthefee** fes the conflru* it ton end, demo* ft ration of the frfl cafe wiH feme t -1 A 4 : . tt / 7 \ \ l- - _« •5. ' . « . i i /: ■ ' ’ > .v. * * \ - N, ' • V - ' / -• ; v.; - i * • o D.iiii. B If I .5\v.1 Jv.Vil v'. r; TUs PropoJ!- man mc- tio»,thoHghe chanically and redis aifoit ie ly do this propofm* •/noli eafie t* 1 r r t is done me- on> noc regardyng ehanicatly. demonftration, hec jet it noprin may extende his co- €i*le' paffe accordyng to the quantitie of lefTclyne geucn, fo feton foote there¬ of in ohe of the ends of the greater lyne geuen,andwith the other foote marke a pointe in the faid greater lineywhich fhall ctittcof from the greater line a line equaU to the lefle.The eafines .of doing wherof may caufe this m-opofition alfo to feeme vnto fome to berathera principlesthena proportion. But to that we haue before in the former propoiitionaunfwered. \ . ■: • . *, v\ ^ ^ ^vbv-..i , ,\ < i rj u > v:. rr,j -; t ^ r, c ri •* r % . l J - i j . * i j y ji qi ax] j l • (W 5 hv'li f . , ; 03 v tJt 'iiOCJ { \ i . SvWgr[l Theorem. The \fPropfimn. ff there be two triangles ,of which twojides oftldone he equal to twojides of the other, eche fide to his correfpondentfide,and loaning aljo on angle of the one equal to one angle of the other , namely, that angle which iscontayned ynder the equall right lines : the bafe aljo of the one fhall be equall to the baje of the other, and the one triangle fhall be equal to the other triangle-, and the other angles re/nayningfhal be equall to the other an~ gles remayning,the one to the other, vnder which are fubt en¬ ded equall fides . .iiiKCt c , ^ ouffofi ofEiitlides Elemefiteh FoLq. Vppofetbat there be two triangles A B C& © E Fyhd* uing t wo fides of the one, namely A Byand A C, equall to two fides of the other , namely ,to © E and T)Fythe one to the other, that isyA Bto(D E,andACto 1 'DFibauyng al * .jo the angle B A C, equall to the angle £©F. Then I fay | that the ha/e alfo BC is equall toy bafe E F: &j the tru |pj jangle A B Cjs equall to the triangle © E F:andy the o * ' tber angles rematnyng are equall to the other angles re* mayningfbconeto the other yonder which are fubtended equall fyder.tb at is,y the angle ABC is equall to the angle © E F, andj the angle ACB is equall to to the angle © £ E.For the triangle A B C ex* aclly agreyng with the triangle © £F, and the point A being put Vpo the point TEpr the right Ime A B Vpon the right line © Ey the point e B alfo pall exaBly agree 'With the pointe E:for float ( by fuppofitionjtbe line A B is equal to the line © E. Jnd the line A B exaBly agreeyng • with the line © Eythe right line alfo A C exaB* ly agreeth 'With the right line T> F, for that(by Juppofitionjthe angle B A Cis equall to the an* gle £© F .And for a] much as the right line A Cisqlfofby j uppofition ) equall to the right line T>F, i her fore the point e C exaBly agreeth With the pointe F. A* game forafmuch as the pointe C exaBly agreeth with the poynte F, and the point B exaBly agreeth with the point E : therefore the bafe B CjJoall txaBly agree with the bafe E F.For if the point S do exaBly agree With the point £, and the point C with the point Fyand the bafe B C. do not exaBly agre Wytb the bafe E F, then two right lines do include a fuperfcies: which (by the io. com on Jentencefis impoffbk.VV berfore the bafe B C exaBly agreeth the bafe E F, and therforeis equallvnto it. VFherfore thewhole triangle ABC exaBly a* greet h with the whole triangle © EFy<zsr tberfore(by the 8, common Jentence ) is equall Vnto it. And (by the fame) the other angles remay ning exaBly agree With the other angles remay ningyind are equall the one to the other, that is, the angle A BC to the Angle © E Fy and the angle ACB to the angle © F £t If therfore there be two triangles, of which two fides of the oney he equall to two fydes of the other, eche to his correfpondent fide, and hailing alfo one angle of the one equall to one angle of the other } namely, that angle which is ckntayntd y>n* der the. equall right lines: the bafe alfo of the one pall be equall to the bafe of the other, and the one triangle Jhall be equal to the other triangle, and the other an* gles remainyng f hall be equall to the other angles remay mngy the one to the a* ther fender which are fubtended equall fydes: whiche thing was required to be demonfirated. This Proportion which is aThcorerae^hath two things geuen,* namely,the . E.i. equality Demonstration, leading io an abfirditie. T tvs t htngts »en in this post go ft son. r&w thingn e<laa^ty of two fides of the one triangle, to two fides of the other triangle^ and retired in it . the equaiitie of two angles contayncd vnder the equall fydes, In it alfo arc thre thinges required.Thc equality ofbafe to bafe: the equality of field to field: and the equality of the other angles of the one triangle to the other angles of theo* ther triangle, vnder which are fubtended equall fides, * • ' . Hove one fide it r\ r r r t r equniito an t- u ne Ime ol a play nc figure is equall to an other, and fo generally one right ther, & fo gene iync is equall to an other , when the one being applied to the other, theyrex- r*fitfahe- £rearnes agree together . For otherwife euery righte line applied to any right dtTJno- lyne,agreech therwith:but equall right lines only,agree in the extremes. ther, Hom one reniih ^nc re'^rl^nc^ angle is equall to an other reftilined angle, when one of the wed angle is e~ " fides which comprehendeth theoneangle,beingfetvpon oneof the Tides which qu*t to an other comprehendeth the ocher angle, the other fide of the oneagreeth with the other fyde ofthe other. And that angle is thegreatcr,whofe fyde falleth without: and that the lefle, whole fydefalieth within. Whj this parti* ele,ech to hts correfpondent fidefs put. Horn one triaH - gtc is equall to -an other . What the fields er area of a trt- angle ss , and fo vfanjrcBilined figure. What the cir¬ cuits or copajfe af a triangle is, and (o alfo of a- nj reHilmed fi¬ gure. Where as in this propofition is put this particle echetohis correfpondent fide, (ia ftede wherof often times afterward is vfed this phrafe^ one to the other jit is ofne- ceffity fo put. For otherwife two fydes ofone triangle added together,may bee- quail to two fydes ofan other triangleadded together,5 and the" angles allocon- tayned vnder the equall fydes may be equall: and yetthe two triangles may not- withltandingbe vnequall* Where-note that a triangle is faydtobe equall toaa other triangle, when the field or area of rheone is equall to the areaof the others And the area of a triangle, is that fpace, which iscontayned within the fydes ofa triangle, Andchecircuiteorcompafleofatriangleisalinccompofed of all the fides ofa triangle. And fo may you think ofali other reftilined figures. And now to proue that there may be two triangles, two fydes ofone of which being added together, may be equall to two fy des ofthe other £> „ded together, and the angles contained vnderthe equall fydes may be equall, and yet notwithftanding the two triangles vnequall, Suppofe that there be two rcftangle triangles: namely, A B C,andD £ Fvandlet their right angles be B A CandE D F. And in the tri¬ angle A B C let the fyde A B beg. and the fyde A C 4, which both added toge¬ ther make 7. And in the triagle D E F,let the fide DEbe2.and the fide D F foe 5* whichc added toge¬ ther make al fo 7*8do the B fy des ofthe one triangle added together,are equall to the fides of the other ?m# pdeadded together, Y etare both the triangles vnequall,and alfo their bafts. For the area ofthe triangle A B C is 6 and his bafe is 5. And the area ofthe triangle D £ F is V- and his bafettfp 2.9. So that to haue the areas oft wo triangles co be c- quali, it is requifite that all the fydes of the two mangles be equall, eche tohys correfpondent fyde. It happenethalfo fometymes in triangles, that the areas of them bey ugcqua.ll, their fydes added together fhall be vnequall. Andconccan- ofEuclides Elements. Foil fa wife^hcir Tides beyng equall,thrir areas be vnequalUs in theCe figures hcrepuc it is plaine to fee* In the firft * ^ example the areas of the two triangles arc equal, for they are eche i2»and the fides inech ad¬ ded together are vnequall, for in the one triangle the fides ad¬ ded together make 18. and in the other they make 16. But in the fccondexaple the areas of the two triagles are vnequal, for the one is i2*and th’other is l§*Butthe fides added tos gethcr in eche are equall, for in eche they makeiS. That angle is faid to fub- tend a fide ofa triagle, which is placed dire&ly oppofite, 8c againfi: that fide.That fide al fo is fay d to fubtend an an¬ gle, which is oppofite to the angle. For eucry angle ofa triangle is contaynedof two fy des of the triangle,and is fubtended to the third fide* This is thefirfi: Propofition in which is vfed a demonft ration leading to an Thisfrofofatien abfurdicie,oran impoffibilitie,Vyhich is a demonftration that proueth not di- tr,uefjky* redly the thing emended, by principles, or by thinges before proued by thefe hZgtolTaifar principles:but proueth thecontrary therof to be impoffiblc,&: fo doth indirect- dity, iy proue the thing emended, A* % AL — t — 4 — 5 X — , — i * Hti® an angle is fayd to fainted a fade sand a fade an angle. \ i. k ty&The i.T heoreme. Tbej.Trofo/ttion. (tJn lfofcelesjr triangle of two equal ' fide sjhath his angles at the bafe equall the one to the other. <iAnd thofe equal fides be- ing produced jhe angles which are Under the bafe are alfo e* quail the one to the other. . * ^ • mVppofetbdt ABC lea triangle ‘of tn>o equall fy des called Ifofcelesy hauing the ! [fade A B equall to the fide A C . And } (by the fecond peticio)produce the lines A 'Bo* AC directly toy points (DoEpThe I fay, that y angle ABC is equal to the angle ACB:andy ji angle C B 2) is equal to tj? angle B C£,Tdke in the line B£)a point at all aduentures, and let the fame J w * V / fTbejirBcBookg fame be F }and(by the third propofttion)from the greater line, namely % AE9 cut of a line equall to A F being the lefle lineyand let the fame be A G : and draw a right line fro the pfmt F t0 l!je Pomt C>af1^ an 0t^er from the point G to the point BJN.OW then for as muche as AF is equall to A G^and A B is equall to A C, therefore thefe tWo lines F A and A C are equall to thefe two lines G A and ABfthe one to the other , and they containe a common angle, n am ely,that which is co* tained Vnder FA G: w her fore (by the fourth pro » poftion)the bafe F C is equall to the bafe G <B: and the triangle A F C is equall to the triangle A G Byand the other angles rental* ning.are equall to the other angles remaining the one to the other yVnder which are fubtended equall fdes : that is,tbe angle ACFis equall to the angle ABG , and the angle AFC is equall to the angle A G (B, And for af much as the whole line A F is equall to the whole line A Gyof which the line A ‘B is equal toy lyne A C, tberfore the ref due of the line A F ^namely ythe line B Ffs equal to the re * fidueofthe line AG ^namely, to the UneCG (by the third common fentence ) And it is proued that C Fis equal to B G . Not* therfore thefe tWo BF&FC are equall to thefe two CG and G B the one to the other, and the angle B FCis equall to the angle C G B,and they haue one bafe, namely, B Cycommon to them both: w her fore (by the 4. propofttion) the triangle B FCis equall to tin trU angle CGBy and the other angles remaynyng are equall to the other an* gles remaining eche to other y Vnder which are jubt ended equall fides.Wher* fore the angle F BC is equall to the angle GCBy and t he angle B C Fis equall to the angle CBG . ISloW forafnuch as the whole angle ABG is equall to the “whole angle AC F( as it ha th bene protied)of which the angle CBG is equal to the angle B CF: therfore the angle remayning: namely ,A B C is equall to the angle remaining3namelyyto A C B(by the third common fentence) And they ar the angles at the bafe of the triangle A B C.a nd it is proued that the angle FB Cis equall to the angle G C Byand they are angles Vnder the bafe . W her fore a triangle of two equall (1 ides hath his angles at the bafe equall the one toy other # And thofe equall Jides being producedy the angles which are Vnder the bafe art alfo equall the one to the other : which Was required to be prou ed. for that of EuclidesElementes. Fd.\ fides of the triangles A FC and AGB& alfo thefy des of the triagles B F C SC C G B run fo one withinan other,thcrforc I hauc here put the diftin&iy,name* ly,the triangles F A C and B t C on one fyde of the figure of the propofitio SC the triangles A GB and C G B on the other fydc.-fo thatyou may with Idle la¬ bor fee the demonftracionplayncly, ir That tn an Ifofceles triangle,thetwo angles aboue the bafe arcequall,may Cuherwife be demonftrated without drawing lines beneath the bafefomwhac ab tering the conftru&ion.Namely, drawing the lines within the triangle jwhiche before were without it after this manner. Suppofethat^-ffCbe an Ifofceles trianglc:andinthclinc A B take a point at all aduentures.and let the fame be D , And from the line AC cut of a line eqnalltothe line AD. Which let be A £,And draw thefe right lines B E,D C, and'D E , Now forafmuch as in the triangles,^ B E, znd AC D> the fifie AB is equall to the fide A C, by fuppofition , and the fides A D and A E are alfo equall by conftrudion,and the angle at the poynt A is common to them both : therfore.by the fourth propofiti- on,the bale B Eis equall to the bafe D C. And,by the fame, the angles remayningofthe one triangle a re equall to the angle s remay ning of the other triangle. Wherefore the angle A B Eis equal to the angle A CD.Againe forafmuch as in the triangles B 'D £,and CED the fide DBis equall to the fide £ C, and the fide B E to the fide D C, and the angle DB E is equal to the an¬ gle £ C D, and the bafe D E being common to both triangles is equall to it felfe: there¬ fore the angles remayningofthe one triangle, are equall to the anjgles remayningof the other triangle. Wherfore the angle ED Bis equall to the angle D EC: 8c the angle DEB is equal to the angle £ D C.And forafmuch as the angle E D B is equal to the an gle DEC, fro which are taken away equall angles D E B,& E D C , therfore by the co- mon fentence the angles remayning,namely,B D C and CEB are equall; And as it was before manifeft the fides B D and D Care equall to the fides C E and £ B the one to the other,thatis,echtohiscorrefpondentfide : and the bafe BCis common to both the triangles : wherfore the angles remayning are equall to the angles remayning the one to the other,vnder which are fubteded equall fides. Wherfore the angle D B Cis equall to the angle £ C B. For the line D C fubtendeth the angle D B C,and the line £ B fubte- deth the angle £ C2:which two lines are as we haue before proued equall. Wherfore in an Ifofccics triangle,the angles at the bafe are equall , though the right lines be not produced. To prouethis alfo, there is an other dcmonflration of Pappus much fhor* ter which needeth no kindofaddition ofany thing at all:as followeth. Suppofe that ABC be an Ifofceles triangle ,& let the fide A B be equall to the fide zA C.Now then vnderftand this one triangle to be as it were two triangles. And thus reafon • For¬ afmuch as in the two triangles A B C and A CB,<tA B is equal to AC & AC to A £,therfore two fides of the one are equall to two fides of the other, ech to his correfpondentfide, & the angle Cis equall to the angle C./££, for it is one and the felfefameangle.Wherforeby the 4.propofition the bafe CB is equal! to the bafe B C , and the triangle zA B C is equall to the triangle A CB', and the angle AB Cis equall to the angle q ACB, and the angle AC B to the angle AB C.-for vnder them ^ are fubtended equall fides ,namely, the lines AB Sc A C. Wher fore in an Ifofceles triangle,the angels at the bafe are equall. E,iH, Tlx<? An ethtr dema- ft ration tnuen- ted by Proelut . An ttkerdemf* ftration inuen- tedby Pnppnt. Mtlepus •the invent or of thit propofition. Demonffration leading to an dmpojfibthtj^ ¥:6i thief efi and tn ift proper kind tsfczmterfon., .\c.-wp \ ,■ % s'« TbefirJb'Bookg The old Philofophcr Thales Milefius was thefirft inuenrer of this fifth propofiti- on,as alfo of many other. ^tefThe third Theoreme . 'The fixtTmpoJitwn. If a triangle haue two angles equall the one to the other : the fdes alfo of the famejtohich fubtend the equall angles 7(halbe equall the one to the other . ppofe that ABC he a triangle , hauing the angle ABC equall to the angle AC 3. Thenlfay tbattbejUe AB i ^ f — " * / 7 - J - I h equall to the fide A C.For if the fide A B he not equal \to tbejide A C3tben one of them is greater .Let A 3 he ■the greater, And by the third propofticn Ofom A B he* fing the greater cut of a line equal to the lefie line fvhich Cj\h AC And let the fame be3)B tAnd draTbe a line front thepoynt 3) to the poynt C.lSloStf forafmuchas the fde 3) 3 is equall to thefyde A Cyand the line 3 C is common to the both therefore thefe tlbofydes ID Band 3C are equall to thefe Wo fydes AC&> CB tbeonetotbeotber .Andtbe angle 3)3 Ch h fuppofytion equall to the angle AC 3. VVher • fore( by t he 4 proportion ) the hafe 3)Ch equall to the hafe A 3 :&(by the fame)the triangle !DB C is equall to the triangle AC 3:namelyy thelefe triangle Vnto the greater triagleywhicb is impofi hle.W before the fyde A Bis not Unequal to the fide AC . VVher fore it he* qual.If ther fore a triangle haue Wo angles equall the one to the other:the fydes alfo of the fameffohich ftihtende the equall angles } fall he equall the one to the other * spbicb Tl>as required to he demon flrated. ’J'.jL ( In Geomecrieis oftentimes vfcdconucrfionofpropofitions.As this propo.' fition is the conuerfeof the propofition next before.The chiefeftandmoftjprOC per kind ofeonuerfion is,when that which was the thing fuppofed in the formpr propofmon}is the conciuilon oftheconuerfeandfecond propofition : and con? trai-fwife that which was concluded in the firft:, is the thing fuppofed in.tHc fe- cond as in the fifth propofition it was fuppofed the two fides ofa triangle to be equal, the thing concluded is, that the two angles at thebafe are equall Sc in this propofition, which is the conuerfe therof is fuppofed that theangles at the bafe be equalftWhich in the former propofition was the conclufion*And the con- ciufion is,that the two fy des fubtending the two angles are equall, which in the former propofition was thefuppofition.This is the chiefeft kind of cpniierfiofi ^oiforraeansUcirayne* 0, T jpg conelufont in the fifth pro¬ pofition. T he fxt prgpo- ftion ts the con-' uerfe as tea¬ ching the ftp tonclufon caefye The conuerje at touching thefe ■» send cotsclufotti jl wfojy* Thereis an other kind ofconuerfion,bur not To full a conuerfion norfo per- Another hind of kSt as the firft is* Which happeneth in compofed propofitions,that is, in fueh, which haue mo luppofttions then onejandpafife from thefe fuppofitions to one conclufion. In theconuerfes of filch propofiti6s, you palfefrom the conclufion of the firft propofition, with one or mo of the fuppofitions of the lame: Sc con¬ clude fome other fuppofition of the felfe firft propofition: of this kinde there are many in Euclide.Therofyou may hauC an example in the S.propofition be¬ ing the conuerle of the fourth.Thisconuerfion is not fo vniforme as the other, but more diuers and vneertaine according to the multitude of the things geuen^ or fuppofitions in the propofition. Bucbecaufe in the fifth propofition there are two conclufions, the firft, that the two angles atthebafe be equall: the fecond^that the angles vnder thebafe are cquall: this is to bcnoted,that this fixe propofition is the conuerfe of the fame fifth as touching the firft conclufion onely . You may in like mancr make a con- uerfe of the fame propofition touching thefecond conclufion therof. And chat after this maner* CV He ttyo fides of a triangle beyng produced \if the angles vnder the bafe be ecjualfthe J aid triangle * jhatl be an Jfofceles triangle Jn which propofmo the fuppofition ls.that the angles vnder the bafe are cquall : which in the fifth propofition was the conclufion: 6c the conclufion in this propofition is.that the two fides ofthe triangle are equal, which in the fife propofition was the fuppolition.Butnowfor proofeof the laid propofition: Suppofe that there be a triangle AB C, & let the fides s^^andix/C* be produced to the poyntesU and G , and let the angles vnder the bafe be cquall, namely,the angles D B C,and GCB . Then I fay that the triangle ABC is an Jfofceles triangle . For take ii* the line AD a point which let be E, And vnto the line BE put the line CF equallfby the 3 .propofitio ^).And draw thefe lines E C,B F,and E F. Now forafmuch as BE is equall to C F, and BC is common to the both , therfore thefe two lines B E&B C*,are equall to thefe two lines C F and C B the one to th e other, & the an¬ gle E B Cis equall to the angle FCF by fuppofition. Wherfore(by the 4,propofition ) the bafe of the one is equall to the bafe of the other,and the one triangle is equal to the other trian gle , & the other angles re- mayning are equal vnto the other angles remayning, theone to the other,vnder which are fubtended equall fides, Wherfore the bafe ACis equall to the bafe FF,and the angle B EC to the angle CEB , and the angle CB Ft o the angle B C F.Bqt the whole angle EBCis equal! to the whole angle F CB , of which the angle F B Cis equall to the angle ECB : wherefore the angle remayning EB Fisc-* quail to the angle remayning F C F.But the line B Eis equall to the line C F,Sc the line B F to the line CE} and they contayne equall angles: wherfore by the fame fourth pro¬ pofition the angle B E Fis eq uall to the angle C F F. Wherfore by this fixt propofition the fide <iA Eis equall to the fide A F: of whiche B E is equall to C F, by conftru&ion : wherforefby the third common fentence) the refidue A B is equall to the refidue A C Wherfore the triangle A B Cis an Jfofceles triangle. if therfore the two fides of a trian¬ gle being produced, the angles vnder the bafe be equall,the fayd triangle fhall be an /- fifieUs triangle:which was required to be proued. This moreouer is to be noted, that in this propofition there may be an other cafe* a* other cafe its for in taking an equall line to eA Cfrom A B, you may take it from the poynte cA and thufxtpropof. E»iiij. not < tnn. iionJiruiiioffa Demtnpiratiotii Thefirft'Bookp Mvt from the poynt , And yet though this fuppbfition aifo be put the felfe fame abfurdity will follow. For fuppofe that A C be equall to A D :aiid produce the line CiA to the poynt£: and put the line AE equal! to the line TO Z?(by the third propolitionjwherefore the whole line C E is equall to the whole line AB ( by the fe- cond common fentece ) Draw a line from the poynt-E to the point jB.Andforafnuich as the lineaxf B is equall to the line £C, and the line's Cis common to them both, and the angle <sA CB is fuppofed to be equall to the an¬ gle SC: Wherfore (by the fourth propofition ) the tri¬ angle EBCis equall to the triangle AB C, namelye,the whole to the part: which is impoffible. t Qtman’lrdt'tcn le*dmg to An abjnriittfe'. The 4.. Theoreme. The 7. Tropofition - fffrom the endes of one line, be drawn two right lynes to any pomteithere can not fro the felffame endes on the fame fide, be drawn two other lines equal to the two firjl lines , the one to the other gonto any other point. Or if it be pofitble: then from the endes of one & the felf fame right line , namely, A <3, from thepointesf I fay) A and 3, let there be drawn tspo right lines A. Cand C3to the point C; and from the fame endes of the line A 3, let there be dr assert tSbo other right right lines A3) and 3) 3 equall to the lines A Cand C 3 the one to the other # _ _ _ js,eche to his eorrefpondent fme^and on one and the fame fide, and to an other point e 3 namely , to3):fo that let C A be equall to 3) A beyng both drawen from one end,that is,A:& let C3be equall to 3) 3,be» yng hothalfo draw from one ende,that isfi.And (by the firjl peticion) draSP a right line from the point Cto the point 3).Kow for aj much as A Cis e* qual to A 3),the angle A C3)alfo is(by the i.pro* pofition)e quail to the angle A3)C: Therfore the angle A C3) is lefie the the angle 3 3) C, Wher* f ore the angle 3 C 3) is much le fie then the angle 3 3) C. Againe forafmuch as 3 C is equall to 3 3), and tier fore dlfo the angle 3 C D is equall to the angle 3 3) CtAnd it is proued that it is much lefie then it: which is im pofitble Jf tber fore from the endes of one line , be drawen two right lines to any point et there can not from the felfe fame endes on the fame fide be drawn ftoo other lines equall to the two firjl lines , the one to the other jvnto a* ny other point: Which ip as required to be demonfir at ed^ In of Eudides Elements . FoLij . In this propofition the conclufionisa negation, which very rarely happe- Negative and* neth in the mathematical! artes , For they euer for the moft part vfc to conclude affirmaciuely,&: not negatiuely . bora propofitio vniuerfallalfirmatine is moft uLttartgt!™ agreahlc to lciences,as faith AnDotle^nd isofit felfeftrong, and nedeth no nega- due to his proofe.Butan vniuerfall propofition negatiue muftof necefiitie haue to his proofe anaffirmatiue,For of oncly negatiue propofitions there canbeno deniormrations.Aiid therfore fciences vfingdemonftracion, conclude affirraa- ciuely^and very feldome vfe negatiue conclufions. S&dJn other demonjlration after Campanus . Suppofe that there be a line A B , from whofe ends A a ndl?, let there be drawen two lines A C and B C on one fide , which let concur in the poynt C.Then I fay that on the fame fide there can¬ not be drawen two other lines/rom the endesof the line -4 2?, which Hull concur at any other poynt, fo that that which is drawe from the point A fhall be equall to the line A C,and that which is drawen from the point 3 flialbe equall to the line BC. For if it be poflible,let there be drawn two other lines on the felfe fame fide, which let concurrent the point D,and let the line AD be equall to the line ACy8c the Dtuers carts line B D equall to the line B C. Wherfore the poynt D fhall fall either within the trian- thts gems„j}r^ gle ^ 5 C,orwithout,For it cannot fall in one ofhhefides,for then a parte fiiouldbe e- tl0„, quail to his whole. If therfore it fall without: then either one of the lines A D and D B fhall cut one of the lines A C and Chords neither fhall cut neyther .Firfte let one cut firfieafe. the other and draw a right line from CtoD. Now forafmuch as in the triangle ACDr thetwofides^4Cand^4DareequaIl,therforetheangle <•// CD is equall to the angle A DC}by the fifth propofitio: likewifc forafmuch as inthetriagle5Ci),thetwo fide* 2? Cand B D are equall,therfore by the fame, the angles BCD ScBDC are alfo equall. An'4 forafmuch as the angle B DC is greter the the angle ADC , F it followeth that the angle B CD is greater then the angle ACD , namely ^the part greater then the whole -.which is impolTible. But if the point D fal without the triangle ABC Jo that the lines cut not the one the other,draw a line from D to C. And produce the lines B D8c BC beyond the bafe CD, vnto the points E8cF. And forafmuch as the lines A C and AD 2 re equall, the angles A C D zndA D C fhall alfo be equall, by the fifth propofition : like- wife for afmuch as the lines B Cand B D are equal, the angles vn- der the bafe,namely,the angles F D C and E CD are equall , by the feconde part of the fame propofition . And for as much as the angle ECDis lefle then the angle ACD : It followeth thattheangle FDCislefie the the angle ADCi which is impoffible.’for that the angle A DCis a part of the angle F D C. And the fame inconuenience will follow if the poynt D fall within the triangle <lA B C, c p > SbfThe, fift Theoreme. "The 8. Trnpofition* Jftxto triangles haue two fdesoftloone equall to two fides of the other ,ecbe to his correspondent fide haue alfo the bafe of the one equall to the bafe of the other : they fhall haue alfo the angle contained vnder the equall right lines of the one,e-» quail to the angle contayned vnder the equall right lynes of the other. Fa* Suppofe sv A Demonftrarion 'teAding to an | omfofjlbthtj , •>\j Tbefir'sl'Booke V ppofe that there he two triangles A B C and V E F:& let thefe two /ides of the one AB and A CJbe equal! to theft two (ides of the other ID E,and D Fyechto his con ref pendent fide. that is ^A BtoV E, and AC Jo D F.Zsr 'let the bafe of the one jiamely ,B C be equal to the bafe of the other .namely jo E F.Then lfay jbat the angle (BA ,C is e quail to the angle EDF.For the triangle ABC ex* aftly agreing With the triangle DE F.and the- point B being put Vpon the point E^and the right line BCvpon the right line B F : the point C Jhall exactly agree la ith the point F (for the line B Cis equal l to the line EF) And B C exaftly agreeing With E F the lines alfo B A and A C frail ex* aftly agree With the lines E D&D F4 For if the baft B C do exaftly a* gree With the bafe F F. but the (ides \B A fjr A C doo not exaftly agreed the (ides E 2) & DEJbut differ as F G esrG Fdce. the frornji endes of one r j dyne fhalbe drawn two fight lines to a poynt,& from the felf fame endes on the fame fide (halbe drah?h two other lines.equal to the two fir (l lines. y one to the o« ther„and vnto an other poyntibut that is impofttblefby the feuenth propofitio) V F her fire the bafe B C exaftly agreeing with the bafe EFjbe (ides alft BA and AC do exaftly agre With the (ides ED and D F.W her fore alfo the angle B .AC Jhall exaftly agre lb the angle E D Fyand therfore (hall alfo be equal to it. If the / fore two triangles bane two (ides of the one equallto two fide s of the othenyech to his cor refpbndent fideyand haue alfo the baft off one equallto- the haft -vf the other.they j hall haue alfo theanglecantayitedwdierthe eqrnll right lines of the dne/qUaUto the angle contayhed Vnder thefquall right linesofthe othenwhich Was re c : 'H* f\ r“> *r\ . • i ; Tft i s T heorcme is theconuerfe of tfre fourth , biiti'cisflOtthe chiefeftand ,/x This proposition is the conuer fe of the fourth, but not the chts feji hnd of con- nerfiott. ■pofttio .Whole conu erfe t h is is ,i s a copound rheorcm^hariingt^o tb«%$ gcue or iqpp.o fed, which arethefe: the one,tharaWq fides of the one triagle be equal to t wo Mes of the other frugie: th’othefjthat the anglecbtaintfd bf the tWo tides of ttfo^fs'cqUalrothe-angfeoo'nta_ined'oi'Chctwoii’des^.t’h'one:buclwth^foon- geft mhes one thing required, whiche is, that thebafe^of the one,is equal to the bale' of the other* No^ th this S^propofftio.bdhg the bafeHf.tlie one is equal to phebafe ofth’other.is thefuppofitionsorthe thing ges ■ ne; which in the former propofitio was the conclufib. And this, that two Odes of ft Fobs. tkeonc areeqqaH to two fidesof the other, is' in this proportion alfo afiip^ pofkion i, like as it was m the former propofitionifo that it is a thing geuen ineis ther proportion. The conclufionofthis proportion is that the angle enclo fed of the two equal! fidesof the one triangle is equalltothe angle enclofed of the two equall fides of the other triangle: which in the former propofiuon was one of tbe things geuen. Philo and hisfcholas demonftrate this proportion without thehclpe of the former propefition,in this maner. •■'4 Suppofe that there be tw© triangles A.B. C and D E F, hauing two fydes of the one equall to two Cycles of the other7namely,v4 B and A C equall to D E and D F , the one to the other, & the bafe B C equal to the bafe E F . And for that the bafe 2? Cis equall to the bafe E F,therfore the one being applied to the Other they agree Place the two triagles ABCScDEFi n one & the felf fame plaine fuperficies.,& apply the bafe of the one to the bafe of the other :But yet fo that the triagle A B C be fet one the other fide of the right line £ F,that the top of the one may be oppofite to the top of the other . And in {lead of the triangle AS C put the triangle E F G as in the figure. And let D E be equall to E <7, and F) F to F G. Nowe then by this meanes (hall hap¬ pen diners cafes ♦ For the line F G may fall diredtly vpo the line D F,orit may fo fall that it may make with the line 15 F an angle within the figurejor with out. , . ' j / * . 1 - =1 ? Firft let it fall dire&lye . And fqraf- mqche as the line D E is equall to the line E 6’,anclD FG, is one rightc line: therfore DEG is an Ifofcelcs triagle: and fo,by the fifth pt-opofition,the an¬ gle at the point D is equal to the angle at the poynt G : which was required to beproued. But ifit fall not dire&ly , but make with the lineD F an angle within the figure, drawe a line from D to G. Now forafmuch as E D and E G are equall ,and the line D G is the bafe’therfore by the fifth propofitio, the an¬ gle E D G is equall to the angle E G D . Agayrte foraf¬ much as D F is equall to F G,and;D G is the bafe : ther- fore by the fame,the angle F D G is equal to the F GD: and it was proued that the angle E D G is equall to the angle E G Diwherfore the whole angle ED F is equal to the whole angle FGB : whione was required to bepio- ueeb yin other detitt* firation itutend ted bj Philo. But if the line FG make with the line D F an angle without the figure.-draw a right line without the figure V.ij. from After this de~ monftrsstio thre safes in this fr& fitters* Thefirft cafiti T he fieetsd cafiti The third cafi* from thepoyirt D to the poynt G. And forafmuch as D E and E G are equall, and DG is the bafe,therfore by the fifth propofitioiuhe angles E D G and D G E are cquall.A- -gaine forafmuch as D F is equall to F G,and D G is the bale, therfore,by the fame, the angle F D G is equal! to the angle F G D . And it was proued that the whole angles E 3>G & D G E are equall theonc to the other: wherfore the angles remayning E D F & EG F are equall the one to the other,which was required to be proued. i 5 ZmBrn&im* DetMOu&ration. c> " J ‘ ' - -• - - i i 1 ‘ l'.* ^&The ^.fP roblcme. T he p.Propojitton, sot EuA ni ?:;■?, ' : nsttt To deulde a rcBiline angle geuenfinto trn equal l partes, v- t ■ ■ ‘(y - . ... ri , . *. . . . . ■ , V ppofe that the reftiUne angle geuen he B A C It is re* qutredto deuide the angle IB A C into two equal partes. In the line A B take a point at ad aduentures, let the fame he D.And(by the third proportion) from the lyne 1 A C cutte of the line A E equall to A ID, And (by the firft petition) drast aright line from thepoint fDto the _ _ _ ! point E.Andfhy the fir ft propofitm)vpon the line ID S' defcribe an equilater triangle and let the fame be DFEy and(by the fir ft peti • cion) draSoe a right line from the poynte A to the point F. Then l fay that the angle BAG is by y line AF deuided into two equal partes. For for* afmuchas A V is equall to A Ey and A F is canton, to them hotktherfore thefe noo D A and AF^are equall to thefe two FA and A Fythe one to the o* ther!But(hy thefirftpropopfm)the bafe D Fis equall to the bafe EF‘Merfdre(hy the ft.propo* /ition)the angle D AF is equal to the angle FAE. & Wherfore the reBiline angle geuen y namely yB AC is deuide d into tioo equal partes by the right lint A F\ Which ms requU red to be done . i . - ^ . In this propofition is not taught to deuidc a right lined angle into mo partes then tw. albeit to demde anangle/o it be a right angle, into three partes, it it L ‘ >. not ofEmlidesElementes. FoLip. It is tmpof doc hard. And it is taught of Vitellio in liis firft bokc ofPerfpe£h'ue,tbe 2S,Propo- fileta deuide fition.Por to deuide an acuteangle into threeequal partes ,is(as faith Proclus)\m- *2-1CHtere‘, ■ poffible:vnlesttbe by the helpc^orothcr lines which are ora mixt nature, Which to three equail thing Tfjcomedes did hy 'fuch lines which are called Concoides linear ho fir ft ferched partes without out the inuemion,nacurc,&: properties, of fuch lines. And others did itby other are meanes:as by the helpe of quadrant lines indented by Hippies & jy/«*w/^«tOther$ 0Ja by Helices or Spiral lines indented of Archimedes .B ut thefe are things of much dif- tme. ficulty and hardnes,and nothere to be infreated oft Hereagainft this propofition may ofthe aduerfary be brought an -Jfinftance. *j„-tnfiaKce ;t For he may cauill that the hed ofthe equ’i'latertriangle (hall not fall betwene the an obieftion or a two right lines ,but in one ofthem^or w ithout them, both. As for example. doubtjvherbj u 1 letted or trots- Suppofe that the angle to be deuided into two equail partes be B A C , and in the line *B <tA take A the poynt D , and vnto the line © A put the line l A E equalfby the third propofition . ) And draw a line fro D to E, And vpon the line DE defcribe(by the firft) an equilater triangle, which let be DEE. Now then if it be poffible that the point F do not fal betwene thelines ABScA C,then it ihal fal e- ther in the \it\eAB or AC, or without them both. Suppofe that the point F be fall vpon line AB, fo that let D F E be an equilater triangle.? Wherfore the line D F is equal to the line F E: & the angles at the bafe are equail, namely, the angles EDF and D E F. Wherefore the whole angle D E (7 is greater then the angle EDF. Againe foraftnuch as zA D is equal! to A £, therefore AD E is an Ifofceles triangle. Wherefore ( by the fifth propofition ) the angles vn¬ der the bafe are equall.Wherfore the angle D E Cis equail to the angle ED B . But it was alfo greatertwhich is impoffible. Wherfore the top ofthe equilater triangle canot be in the right line A 2?. And in like fort alfo may we proue that it canot be in the right Vine A C.Wherforefuppofethat it be without them both, if it be poffible. Andforaf* much as D F is equal to F E,the angles at the bafe are equal,namely,the angles DEF & E D F. Wherfore the angle D E F is greater then the angle E'DV -Wherfore the angle D EC is much greater then the angle E D F.But it is alfo equal vnto it. For they are angles vnder the bafe DE of the Ifofceles triangle AD E. Which is- impoffible . Wherfore the poynt F (hall not fall without the two right lines on that fide . And in like forte may we proue that it (hall not fall without them on the other fide . Wherfore it fhali of neceffi- ty fall betwene them : which was required to be proued* hied the con- fins ft/on , or de¬ mon ftrat son, contajnsth an 9 nruth , and an impoffibihtj: and ther fore it mvft of nece fifty be anfrered \>n~ to, and the fill fe- hode thereof made manifefi. Theremayalfo in this propofition bedmers cafes. (fit fo happen that there Diners cafes in be no fpace vnder the bafe D E to deferibe an equilater triangle, but that of necef- this propofition, fitie;youmuft deferibe it on the fame fide that the lines AB and A C are. For thenthefides ofthe equftater triangle either exa&ly agree with the lines AD and A H, if the faid lines A'D and AE beequall with the bafe D E. Or they fail without thera'jifthe lines A D and A E beleffe then the bafe DE- Or they fall within them, if the faid lines be greater then the bafe D E. > ’ y r -s:n: o- rmn ■ " a ::r : •- , Firft let them exactly agree, And let© A E be an equilater triangle. And in the fide The firft cafe, A D take the poynt G , And from the fide A E cut of a line equal to the line A G( by the third propofition Jwhich let be A fi, And draw thefe right lines (j E,HD and GH, and AF.Now forafmuch as AD is equal to AEfzndA G vnto A /ifttherfore thefe two lines F-iift DAmd sjn he peon d cafe. TbejirB'Boofy A and A Hare equall to thefe two lines £.4 and A Grand they contaync one and thefelfe fame angle, Wherfore by the fourth propofi- tion,the angle G D H is equal to the angle H E G. And the bafe ‘D H is equall to the bafe E G. But the line D G is equal to the line E H:wher fore againe by the fourth propofition , the an¬ gle EGHr equall to the angle 2) HG. Wher¬ fore by the fixt propofition, thebafeGFise- qual to the bafe H F And forafinuch as A His equall to A G,and A F is connnonjto the both, and the bafe G F is equall to the bafe H F, ther fore the angleGAF is equall to the angle HA F. Wherefore the angle G AH is deuided into two equal! partes : which was required to be done. B ut if the fides of the equilater triangle fall without the right lines BA&A C,as do the lines F & £ F,the draw a line from F to, A & produce the line F A to the point G. Now forafinuch as the lines D F and F E are equal. Sc the line F A is common to them both, & the ba- fes D A and A E are equall: therfore(by the eight) the angle D F A is equal to the angle E F A. Agatne forafinuch as D F and F E are equall , and F G. is common to them both, and they containc equall angles (as it hath bene proued ) therefore (by the fourth) the bafe D Gis equall to the bafeGE. And forafinuch as A D is equall to A E,and A G is com- ; mon to them both, Therfore(by the eight)the an¬ gle PAG is equall to the angle E A G .wherefore the angle D A E is deuided into two equall partes : 0 Which was required to be done. •S the third cap. T<4 dcaide a. re&tiine angle ■onto two equall farts Mechani- Jtatlj. Butif the fides of the equilater triangle fal with¬ in the right lines B A and A C, as do the lines D F and F E,then againe draw aline from A to F. And forafinuch as D A is eq uall to A E,and A Fis com¬ mon to them both,and the bafe D F is equal to the bafe F E: ther fore the angle D A F isf by the eight) equall to the angle E A F, Wherefore the angle at the point Ais deuided into two equall partes,how foeucr the equilater triangle be placed: which was required to be done. Tbi s is to be noted/that if a man will me¬ chanically or readily, not regatdyng demon'-, ft rati on, denude the forefaid reftilincangle B AC ,and Co any other re<fhlirreanglegeuenwhatfoeii£f, into two equall partes3he {hall needeonely with one o£ peaing of the compafle tafcen at all adtientures to markc the two pointes D andE, which cult of equal partes of the lines A B and A C,howfoeuerthey happen, and fo ma¬ king the centres the two points Dand E, to deferibe two circles according to the openyng of the compafle .* and fcom the point A to. their interfe&ion, which let be the point F to draw a rightline: which fliall deuide the an¬ gle B A C into two equall partes. Add here note, 'that youfhall not nede to draw the circles all whole^but one* of Euclides Elelnentes . Folio. ly a portion where they cut thbbiib the o'tftttY As m thehgote here in the end of the other fide put. &! H £ Vr V., S-. ... ; . i n r „ •• • ^ ( : ^ ^ The sfProhleme. f The iQ.Tropofition u , VC 3 von-. - :iU T o deuide a right line getten being finite (into tm equall ^ ! "vai 31 partes. V/.i J. nt .3UO 510; Vppofe that the right linegeuenbe A <B .It is required to deuide the line A B into fWo equal partes.Defcribe( by the firjl propo/ition)vp * on the line AS an equila ter triangle and let the fame he A BC. And ^ (by the former propofitiohfdeuide the angle AC B into tyi>o equal l partes by the right line CD.TbenI fay that the i > right line A B isdeuided into two equall partes in the poynt D.For forafnuch as ( by thefrfipropo* ftion)A C is equall to CB , and C D is common to the botktherfore thefe two lines AC <& CD at e e* qua l to thefe two lines BC<jr £D(y one toy other, and the angle AC Dis equaUto the angle BCD. t VVbetf&re(by the ^propofition ) tfy bafe AD is : / • equqll to the bafeBDAVhere fore the righte line JC~' p 3 geuen A B,is deuided into VWoe quail partes in the poynt Dibich 'Was required to be done 4 . ' v Apollonius' 'teacheth to deuide a rightlinebeihgfinite into two eq«aih|)4[|tes after this manner, 1. c r „u .... , . : vjdi* 0* ran *1 /l J a Suppoie that the right line being finite ,, ^ , be AB: which it ii required to deUirle ih?Ji ■ to two equal parts . now the makingthCv .cr; ;• A . vt y centre the point A & the fpaoe A B de- 5 t / Icnbe a circle. Again making the centre , .. . the poynt B & the {pace B A defcribeah 'V“ 1 ' ’A”' orthetdrclsiand froxnths rfc, 5 ' f£^thitthenl^ht find ^ othe^fbrthat they frd the centres to the circumferences of equall circles.And forafmuch as the lines C^AM^ A ‘Dare equall to the lines C B and B D.and the line CP is common to either of them: -V- mis ■••i- \ oUva- ' , l . * v "d 1 -tvi a'-. C on ft ruction* Demon^ratioji, ,nsVtWn\:..i ■ An other way ft deuide a right line being fnit<t, inuented bj A* pellontus. By this way ofdeuidinga rightline, into two equall parts inuented by Apollonius, it is manifcft, that if a man wil mechanically ,or redeIy,not confi- dering the demonftrati5,deuicje the faid rightline, andfo any Tight line geuen whatfoeuer, into two e- quall: partes he nedeonely to marke the poyntsof the rnterfe&ions ofthecirc'les, 8C to draw adinefrS the fa/d interfe&ions, which fhall deuide the right line gcuen into two equall partes : as in the figure X X 5^ The 6.KProblemc. The IvTropoJitim. Vpm a right tme geuen, torayfe tyfrom a poynt geuen in the Jameltne a perpendicular line. : - Vi j V Dpofe that the right line geuen he A '!*>><& let the point in it geuen be Clt is required ftotn the poynte C to rayfe Vp Vnto the right line- A a perpendicular line . Take in the line A C a poynt at all aduentures3 & let the fame beD>and(bythe$ipr.6pojition ) put VntoDC an equall j line C E.And by the firftpropofttionftpon the line D E fteferibe an equilater triangle FID E* ^dra'W a line fro FtoC4 Then I fay that Vnto the right Untgeuen A 'Bgmdfrom the poynte in itgeuen , namely jC is ray/edvp a perpendicular line FC. For fora/much as DChequalto CE the lineC F iscom&n to them botkeherfore thefe ttto DC and C F^are qua l to thefe nto EC ejr C Ffthe one to the other : and(hy the fir ft propofition)the bafeD F is equal to the bafe E F : wherefore (by they tpropo fit ion) the angle DCF is equaUto the angle ECF: and ^ they be fide angles&ut lphe a right line ftanding ypona right line doth make the two fide angles equall the one to the other ^ether of tboje equall angles if {by the.\o,definition)a right angleg^r the line ftanding Tpon the right line is called a perpedicular Ime.VVherfore the angle D C wangle F C E are right angles.VVherfore Vnto the right line geue A fro Cjtxayifedjrp ttotCffcp.bfcb "teas required to be done 4 '■ ■ ' :m3iihlppr h - no > nmoa zi U 3 &oH silt inu..C . ■, ;Unpv.-ifiG?K Although thepoynte, geuen flipuld be fet in one of the endes of the rightc fihe 'jgeufci)»it is' eafy fp tfdt it^s it, Was-heforc ; For producing the line in length iiqpithe poynt by tfie fecorid peticuphjy pfirnay wprkeas you did before. But if one require to ereft ^ r%^t line pcrpendicjqlarly from thepoy nt at the end of tfi e ■ ; iotfisiifipet tr rbidwia. ai gsjiiq Ifcu i|*“* ofEnclides Elementes. FoUu Ivne, without producing the rightlyne, thatalfo may well bee done after thy* raaner, ' ■ ■ • - l i ■ Suppofe that the right line geuen be A 2?,& let the point in it geuen be in one of the endes theroB namely ,in A .And take in the line A B apoint at all aduenturcs,andletthefamebe C. And from the faidpoint raifevp(by the fore Paid propdution)vn- toABa perpendiculer Iine,which let be CEt And (by the 3 .propofition)from the line CE cut of the line C D equal! to the line C A, And ( bythe 9 .Pro- pofitionjdeuide the angle AC D into two equall partes by the line CD, And from the point D raife vp vnto the line CE a perpediculerline,D F,which let concurre with the line C Fin the point F, And drawe a right line from F to A ♦ Then I fay that the angle at the pojnta^ is a right angle. For, foras¬ much as D Cis equall to Ce^f, and CF is common to them both3and they containe e- quall angle§(for the angle at thepoint Cis deuided into two equall partes ) therefore fby the 4. Propofition) the line D F is equall to the line F A, and fo the angle at the point A is equal to the angle at the pointD, But the angle at the point D is a right an¬ gle. Wherfore alfo the angle at the point zsfisa right angle.Wherefore from the point A vnto theline raifed vp a perpendiculer line ^F,withoiitproducingthe line A B . Which was required to be done. F D \ L 3 An other cafe m this proportion* Ctnsimffkn, ; Demonftration, Appollonius teacheth to ray fe vp vnto a line geuenjfrom a point in it geuen, a perpendiculer line^fter this maner* : • v *. .W--C D ' • • : Suppofe that the right line geuen be A B. And let the point initgeue, be C, And in theline a A C,take a point at all ad- uetures,&kt the fame beD. Andfrothelyne CSjCMtofaline equall to the line CD, whiche let be C £.- and makvng the centre D, and the {pace D E, deferibe a circle. Andagaine ma¬ king the centre C,& the fpace E P , deferibe an other circle, and let the point of their interfedion be F, And draw a right line from F to C .Then. I fay that the line F C is creded perpendiculerly vnto the line o 4 B. For drawe thefe lines F D and FF-. which flial by the definition of a circle be either of them equal to theline D E. and thcrforc (by the firfi: common fentcncc) are equall the one to "the - - y > A i / f v ■ \A P\ - C Another Way to ereli a perpf- dictriar Itnc its* vented by Ap» palon'mt. Ctnftruttknp ■ -c . D wjafefj'ssJ wherfore they are right angles, Wherfore the line CF is eroded perpendiculerly vnto theli^^£fromthepointC(4.whichwasrequiredtobedone- - ! v By tltis wav oferedii?ga perpendi^uletiineinuented by Appollonius, it is ilfpmanifc'ft,fhai; if a man y ill mechamcaliy.withoutdemonftration xreci vnto ‘ - * . t ' G-h ‘ Mk Hovtto trtff a ferpendietilatt ttne meehmi* Cexfir&ftian. Ztcmonjlration. TbefirH'Boofy a line geuen from a point gcue in ita perpendi- culeriine: he needeonely on either fide of the pointe geuen, to cut of equall lines : andfo ma¬ king either of the cndes ofthe faid lines (either ofth’endes I fay , which haue not one point cos mon tothemboth) the centres, and thefpace both the lynes added together, or wider then both,or at the left wider the one ofthem,to des fenbethofe portions ofthe circles wherethey cutthe one the other, andfromthepoint ofthe interfc&ion to the point geuen^o drawalync, which {hall be perpendicular rntothe lync ges tie: as in the figure here put it is manifeft to fee« T'heyfProbleme . 7 he iiSPropofition. V Mo a right line geuen being infinite , andfroM a point geuen not being in the fame line Jo draw a perpendicular line • Et the right line geuen he* ing infinite he JBjjr let y point geuen not being in the faid line A (B,be Ct It is re* quiredfrom the point gene, namely, C Jo draVe Vnto the fright line geuen A B,a per * pendiculerline.Take on the a other fyde ofthe line jfB ( namely , on that fide therein is not the pointe C) a pointe at alladuen* tures,and let the fame be S). Und making the centre C,and the /pace C de* fcribe( by the third peticion)a circle, and let the fame heEF Gjohich let cutte the line A B in the pointes E and G.And (by the x.propofition ) deuide the lyne E G into two equal partes in the point ELAnd(by the fir/l peticion)draTV thefe right lines, C G,C H,and C E.Thenl fay, that Vnto the right line geuen A Bjsr from the point geuen not being in it, namely, C, is dralven a perpendiculer lyne C H-Forforafinuch as G His equall to HEy^dAiCis commonto them both: the r fore thefe two fydes G Hand HC,are equall to thefe Ctoo fydes EH&H £ the one to the other: and (by the if definitio)the bafeCG is equal to the baft C Either fore (by the $.propofition)the angleCHGis equall to the angle C HE: and they are fide angles i butis/hen aright line (landing Vpon aright line maketh the tTVofyde angles equall the one to the other, either of thoft equall an* Aes is( by the \ot definition ) a right angle , and the line (landing Vpon the fay de right hne is called a perpendiculer Une.Wherfore Vnto the right line gene A(B , and from the point geuen Cftobicb is not in the line A BfisdraTMi a perpendicu* let fine C H: T&hich as required to be done , Thiy >c x of Enclitics Elementes. Fol.zz. Thi s probleme did Oenopides firft fictde out,confidering the neceflfary yfe thers of to the ftudy of Aflronomy ♦ There are two kindes ofperpendiculer lines: wherofonc is a plaine perpen* diculer lynCjthe other is a folide* A plaine pcrpendiculer line is , when the point from whence the perpendiculer line is drawer), is in the fame plaine fuperficies with the line wherunto it is a perpendicularv A folide perpendiculer line is, whe the point, from whence the perpendiculer is drawne,is onhigh,and without the plaine fupcrficics.So that a plaine perpendiculer line is drawento a right line: 8c afolide perpendiculer line is drawn to a fuperficies* A plaine perpendiculer line caufeth right angles with one oriely line, namely, with that vpon whome it fal- leth.But a folide perpendiculer line caufeth right angles, not only with one line, but with as many lynes as may be drawn in that fuperficies, by the touch cherof* This propofition teachcth to draw a plaine perpendiculer line, for it is drawn to one line, andfuppofed to bc in the fclfe fame plaine fuperficies. f There may be in this propofition an other cafe.For if it be fo, that on the o- ther fide of the line cA B , there be no fpace to take a pointe in butonely on that fide wherein is the point C . Then take fome certaine point in the line s A 2?,which let be D.And making the cen¬ tre the pointC, and the fpace CD,de- feribeapart of the circumference of a circle, which let beD E F.-which let cut the line A Bin the two pointesD and _ F.And deuide the line D F into twoe- A quail partes in the poyntH. And draw thefe lines C D, C H and C F, And for- afmuch as D H is equal to H F,and C H is common to them both, and CD is equall toCFfby the iy.de- finition:)therfore the angles at the point H are equal the one to the o- ther(by the 8, propofition:) &they are fide angles, wherefore they are right angles .Wherfore the line C H is a perpendiculer to theline D _ _ F. But if it happen fo that the circle A which is deferibed do not cutte the lyne, but touche it, then takyng a point without the point E,name- ly,thc point G,and making the centre the poiht C,and the fpace CG,dcfcribe a part of the circumference of a circle: which fhall of neceflkie cut the line AB: andfomay you proceede as you did before. As you fee in the fecond figure. $■&? The 6 . 1‘heoreme. The 1 3. Tropofotion - W hen a right line founding ypon a right line ma^eth any an - gles: thofe ang les jhall be either two right angle s3or equall to two t ight angles* Supfofi Oenopides tha firjl in u enter of this probleme. Two ktndes of perpendiculer lines, namely y<t plaine perpends ■» culer line andst folide . ■' This propoftson teacheth to draw a playne perpendiculer line. *in other cafe m this propofition » ConftuHim^ Demotfrntsossh .\SV,v. : AA» <FbeJir$cBoo%e Vppofe that the right line A B flanging Vppott the right line CD do make thefe angles C BA and ABD. Then l fay, that the angles CBA and AB D are eyther two right angles, or els e quail to two right angles Jf the angle (by the n propoption )Vnto the right line CD, and front the pomtegeuen m it, namely, B,a perpendiculer line B \E. W her fore n Conpruclio#^ . . Demonstration x. definition) the angle C B Eand E BD are right angles , ISloTfr forafmuch as the angle C B Efts equall to thefe Etoo angles CBA and ABE, put the angle EBD common to them both\'it>her fore the angles CBE and E B D,are equal to thefe three angles C B A A B E,and E B D.Agayne forafmuch as the angle DB A is equall vnto thefe two angles D B E and E B A,put the angle A BC common to them both^herfore the angles DBA and A BC,are equal to thefe three angles, DBE, E B A, and ABC. Audit is proued that the an* gles CBE and EBD are equal to the felfefame three angles: but thinges equall to one & the felf fame thing, are alfo(by the firU comma fentence) equall the one to the otbe.VVberfon the angles C B Eand E B D are equall to the angles DBA & A B C, But the angles C B Eand EBD are two right angles^berfore alfo the angles D B A and A BC are equall to two right angles . Wherfore'^hen a right line j landing Vo pon a right line maketh any angles: thofe angles jhalbe either ttoo right angles , or equall to right angle si^hich was required to be demonfirated. An othet demonftration after Telitarius. Suppofe that the right line A B do {land vpon the right line CD. Then I fay,that the jn other Je- tnrQ ang{es J. g C and ABD, are either two right angles,Or equal to two right angles, man$ram»afz Fq„ bfi perpedicuJervntoC2): the is it manifeft, that they are right an glesfby the ter PeiturtfM. conuerfion 0f thc definition JButifit incline towardesthe end C,thenfby the i i.pro- pofitionjfrom the point £,ere<5l vnto thelineCDaperpcndiculerline5£,By whiche conftruftion the propofitio is very manifeft.For forafmuch as the angle A B D is grea¬ ter then the right angle D B E by the angle AB E, and the other angle ABC is leffe then the. right angle C BE by the felre fame angle ABE : if from the greater bee taken away the excelfe, and the fame bee added to the leffe,they Hull be made two right angles. That is,if from the obtufe angle ABD be taken away the angle tAB E, there Ihal remaynethe rightangle‘X>££, And then if the fame angle ABE be added to the^cute an¬ gle CB*A, there {hail bee made the right angle CB E. Wherefore it is manifeft,that the two angles, namely, the obtufe angle oAB D,8c the acute angle <>AB C, are equall to the two right angles CB E and DBE' which was required to be proued. . >1 / / I C FoLip The 7. Theoreme. The 14.. Tropojition • fffvnto a right line* and to a point in the fame linefbe dram Wo right lines, net both on one and the fame fide, making the fide angles equall to Wo right angles : thofe Wo right lynes \e directly one right line. rs:v :^-y= Kto the right line A®,& ^ t-o j> point in it ®, let there be drawn two rightlines ® C,and®®, Vnto contrary .fides, making the fy dean* \gles, namely ,A ® C&A® ®}equall t<r two rbbt an* rrh> r 'VliPti T f/Muth/it ii ri.O'frt C £> x> lines®® and ®C make both one right line. For ifC Band®® do not make both one right line Jet the right line ® E befo drawn to ® Cjhat they both make one right linegfoW forafmuch as the right line A ® / Undeth Vpon the right line C ® Esther fore the angles A® C and A® E are equall to two right angles (by the 1 $ propofition) ®ut (by fuppofition ) the angles A ®C and A®® are equall to two right an? gks:Wherfore the angles OBJ, and A ® E,are equall to the angles C®4,and A®®', takeaway the angle A® C, which is common to them both ,W her fore the angle remay ning A® EJs equall to the angle remaining A® ®y namely , the lefie to the greater which is impofiiblc. Where fore the line ®E is not fo di * reftly drawen to ® Cy that they both make one right line. In like forte may we prone, that no other line , be (ides ® ®, can fo be draWne.VF h erf ore the lines C ® and ® ®,make both one right line, If therfore Vnto a right line,<& to a point in the fame line, be drawn two right lines. pot both on one and the fame fidejna » king the fide angles equall to two right angles: thofe two lines Jhal make dire ft* ty one right line: which Was required to be proued. . An other deraonftrati on after Pelitariiis. Suppofc that there be a right line A B, vnto whofe pointe B, let there be drawen two right lines C B and BD, vnto contrary fides : and let the two angles C 3 A, and D 3 Ay be either two right angles., or equall to two right angles »Then I fay , that the two lines CB and BD , do make dire&ly one right line, namely0CZ>. Forifthey do not,th6 let be fo drawn . vnto CB, that they both make dire&ly one right line Ai CB E: which ihallpaife either aboue the line 2? Dy or vnder it . Firft let it pafTe aboue it.And for as much as j the two angles CB A and ABE3avc( by the former pro- j -E pofition^ equall to two right angles, and are apart of the two .angles, CB A and A3 X>:but the angles C3A and A B D are by (fupp o fi tio n) e q u a 1 1 alfo to two right angles: therefore the parte is equall to the whole which ___ is impoffible. And the like abfurditie will follow if CB ^ G,iii, E leading ta jin alfurdtue. Another (ltd mor.Firatjanaf* ter Pehtarim.r a o pemtnftratioH. 'Thales MilepUi the firf trtn en¬ ter of this pre¬ pays ion. No ean$ruci 'fon in this propof- lion. What hedan- pits are. The eonnerfeof this prepoftio of ter Peis tart as. Thefirjl^oofy E pafle vnder the line 2 D.-namcly^that the whole Ihalbe equall to the parttwhich is alfo inipoflible. Wherefore CD is one right line: which was required to be proued. T he 8. Theoreme. The ijSPropojttion* fftwo right lines cut the one the other: the bed angles Jhal be equal the one to the other* Vppofe that the/e Wo right lints A B and CD, do cat the one the 00 ther in the point Et Then I fay y that the angle A E Cjs equall to the angle IDEE. Eor fora/much as the right line AEJlandeth Vpon the right line l D Cy making thefe angles C E A, and A E D: therefore(hy the 1 3 * propo(itio)the angles C E A, and A E D,are equall to Wo right angles ♦ Agayneforafmucb as the right line D E,flandeth Vpon the right line A B, making thefe angles A E Dyand D E (B:therfore( by the fame propofti • on) the angles A E D,and D E B,are equall to Wo right angler, audit is prouedy thatthe angles CEA,and A ED, arealjo equall to Wo right an* gles. VVherfore the angles CEA,and A E D,are equall to the angles A ED,and D E BtTake a • Way the angle A E D ffchicb is common to them hath. Wherefore the angle remayning CEAyis equall to the angle remayning V E B, And in like fort may it he proued , that the angles C E ByandDE A%are equall the one to the other . If therefore Wo right lines cut the one the other y the hed angles jhalbe equall the one to then* ther : Which Was required to be demonstrated. Thides Mtkfws the Philosopher was the firft inucntcrof this proportion, as but yetit was firft demonftratedby Euclide. And in it there is no confl ruction at all* For the expoiltion ofthe thing gcue,is fufficientinough for thedemonftration* - //^^/^areappofite angles, caufcd ofthe interfe&ion-of two right lines: and are to called,becaufe the heddes of the two angles are ioyned together ia one pointe* The conuerfe of this proportion after Telitarius . Iff after right lines being dr Often from one point, do make fofter angles, offthtch thetfto oppo- fite angles are equall; the two oppofite lines Jhalbe dr often direttly, and make one right line . Suppofe that there be fower right lines A B, A C, A D, and AE, drawen from the poyntA,makingfower angles at the point A: of which let the angle BAG be equall to the angle DAE, and the angle B A D to the angle CAE. Then I fay, thatB E and C D are oneiv two right lines: that is^the two right lines BA and A Bare drawen dire&ly, and 0 mentes • FoLi\. anddco make one right line, andlikewfie the two right lines C A and A D are drawen diredtly ,and do make one right line. For otherwife if it be poiTible , let £ £ be one right line, and iikevvife let C G be one right line. And for- almuch as the right line £ A dandeth vpon the right line C G , therefore the two angles EAC and EA<j> are ( by the 13 proportion ) equall to two right angles. And for- afmuch as the right line G A dandeth vpon the right line £ £: therefore (by the felfe fame ) the two angles EAG &ndF A G,zre alfo equall to two right angles. Wherefore taking away the angle EAG , which is common to them both, .the angle £ A Cdhall ( by the thirde common fen- tence) be equall to the angle £ A G: b ut the angle £ A C is fuppofed to be equall to the angle BAD. Wherefore the angle BAD is equall to the angle FAG , namely a part to the whole: which is impoffible . And the felfe fame abfurditie will follow, on what fide foeuer the lines be drawen. Wherefore B £ is one line,and CD alfo is one line: which was required to be proued. Demouffratim leading to a>» abfurditie. The fame comer fe after T rocks. If vnto a right line , tend to a point thereof he drawen two right lines , not on one and the fame fide , in fitch fort that they make the angles at the toppe equall: thofe right lines Jhalhe draWen dir Ally one to the other , and jhalmake one right line. Suppofe that there be a right line A B,and take a point in in C. And vnto the point in it C, draw thefe two right lines CD and CE vnto contrary fides, making the angles at the hed equal, namely, the angles A CD and B CE. Then I fay, that the fines CD and C E are drawen dire&ly, and do make one right fine . For forafmuCh as the right line CD Handing vpo the right fine A B,d oth make the angles D C A and DC B equall to two right angles (by the 1 3 propofition :)and the angle DC A is equall to the angle £ C£: therefore the angles DCB and B fiE are equal to two rightangles. And forafinuch as vnto a certayne right line £ C,and to a point thereof C,are drawen two right fines not both on one and the fame fide, making the fide angles equall to two rightangles, therefore (by the 1 4, 'propofition j the lines C D and C E are drawen dire&ly, & do make one right line,which was required to be proued. The fame cbm - eterfe after Peli*> tariusgvhtch it demonflratid dirt Illy, The fame may alfo be demonftrated by an argument lea¬ ding to an abfurditie. For if CE be not drawen dirc&ly to C D, fo that they both make one rightline, then (if it bee poffible) let CF bee drawne dire&ly vnto it. So that let D CF be one right line. And forafinuch as the two ri"ht lines AB and £>£docutte the one the other, they make^the hed angles equall (by the 1 5. propofition) Wherefore the an¬ gles A CD and B C Fare equall j but (byfuppofition) the angles A CD and B C S are alfo equall . Wherefore (by the firft common fentencej the angle B CS is equall to the angle B CF: namely,the greater to the lefle : which is im- polfible. Wherefore no other rightline befidtes C£is dra¬ wen dire<% to CD. Wherefore the fines C D andCFare dram* di.re£tly,andmake one right fine; which was requi¬ red to beproued. The fame ct7t* uerfe after Pro* tlut demonflra- tedindirehlj. The fir fl ^Boo^e Ww*Corrol- , Of this fiuetenthPropofition followeth a Corrollary. Where note that a- *??"• CVc/% is a Proportion, whofedcmonfkation dependeth of the demonftration of an other Proportion, and it appeareth fodenly,as it were by chance offering it felfe vnto vs: and therefore is reckoned as lucre orgayne. The Corollary which followeth ofthis propofition, is thus* A Corrollary fi Mowing ofthis progefittan. If fetter right lines cut the one the other: they make fitter angles equall to finer right angles. This Corollary gauc great occafion to finde out that wonderful propofition in* uented of pithagdras, which is thus. A TVonderfull propofitien in- ■uented by Pi- thagoras . Only three kindcs of figures of many angles , namely ,an eqmlater triangle , a right angled figure, ofj ottcrfid.es> and a figure of fixefidesjoaumg e quail fiides and equal angles } canfill the ttholefpate ■about a point gheir angles touching the fame point. . -■ Euery angle ofi an equiUter tri angle is equal to fwo third partes of a right angle Euery angle ofi a fixe angled fi¬ gure is e quail to a right angle, nnd to a third part of a right angle. xj- • •» v* vovV \ - v < \ N *.i V‘> ^ -}. » *. V/ * WiYVU ♦, Y » Y . • lir,' s* V* Euery angle of an equilater triangle contay- neth two third partes of a right angle ; fixe ty roes two thirdes of a right angle make fower nghc an? gles. Wherefore fixe eqmlater triangles fill die whole fpace about a point which is equal to fower right angles,as in the i, figure. Alfo euery angle of are&angle quadrilater figure is a right angle:wher fore fower of them fill the whole fpaceas in the 2. figure*Eue«y angle otafixe angled figure, i's.equal ' to a right angle, and moreouertoathirdparcofa tight angle. But a right angle,and a third part of a right arigkjtake thre times,make 4,right angles: wherefore three eqmlater fixeangled'figufes.fill the whole fpaee about a point: which fpace (by this Corrollary) is equal! to fower right angles: as in the third figure. Any other figure of many fids, howfoeueryou ioyne the together at the angles, flial either want of tower anglesjor exceede them . By this Corrollary alfo it is manifefi that • sf mo then two lines, that is, three* or fower, or how many feeuer do cut the one the other in one point, all the an¬ gles by them made at the point fhalbe equal! , to fower right angles., For they r. fill the place of fower right.angl.es. And it is alfo manyfeft, that the angles by thofe right lines made are double in number to the right lines which eutte the one the other. So that if there be two lines which cut the one the other, the are there made fower angles equal! to fower rightanglcs; but if thre, theta •jt:u ar there made fixe angles ; if fowetjeightanglesjand fp infinitly,Eor euer th« mukicu.de, or number’ of ofthcangles is dubled to the multitude of the rq*hf lines which cut the one the other. And as theaogles increafe in multitude, £0 diminifii Of nuciian jzimemes . FoL 25. dimmififi they is cteuided is alwayes one and thefelfeia'mle'thing;, natndyyfbyer rijghc’aiigtes, ' l . : ’ . u ' cfke 9 SI heoreme. The i6SPropofition. Whenfoeuer in any triangle, the lyne of one fyde is drawn „ , . « > » r . 1 /» 11 * 1 forth in one of the Wo reater vni- Vppofe that 4 ® C be 4 triangle: & let one. of ) fides jfierof , namt* p \ly M C he prqducedMto the point ~ IDfThen I fay, that the outwarde angle A CD }ts greater then any one of J ttoo inward and opposite angles fat is fen the angle CB A,or then the angle B A CJDeuide the line A C (by the io .propofition) into two equall partes fin the point E.Anddra'fr a line from the point B to the point E: And (by the 2. petition) extend BE to the point E.jfnd (by the z, propofition) onto the line B Eput an equall Ime E F. And(by the fir fl petition) dra'toa line from F to C.and (by the z.petici * on) extend the line AC to the point G.MoTo forafmuch as the line A E, is equall to the line E C, and B E is equall to E E: therfore thefe nvo fide s A Eand E B, are equall to thefe neo fides C £ and E F, the one to the other, and the angle ft E Bfis (by the i 5, propofition) equall to the angle F EC, for they are hedan* gles: therefore (by the 4. propofition) the bafeA B is equall to the bafe F C: And the triangle ABEis equall to the triangle F E C:and the other angles re* mayning are equall to the other angles remaymngfe one to the other fnder Tfihicb arefubtended equall fides, VEherefore the angle B AE is equall to the angle E C F„ But the angle E C ID Js greater then the angle E C FyVherefore the angle A C £)}is greater then the angle B A C. In like fort alfo if the line B C be deuidedinto two equall partes , may it be prouedfat the angle B C &3tbat is, the angle A GAD, is greater then the angle ji B C.VVhenfoeuer therfore in any triang ley the line of one fide is dr awen forth in length : the outward angle fbalbe greater then any one o f the ttao inward and oppofite angles : ’tobich l#a$ required to be demonstrated. jin other demonjlration after Telitarius, Suppofe that the triangle geuen be A *B C, Whofe fide AB let be produced vnto H*j* the Cgnftr&aim. Demsnftrdtiem. An other D&* wonfiratian af-* ter PslitHriut, tire point D. Then I fay, that the angle D B Ck greater then either of the angles BAG an dA CB* For forafmuch as the two lines, A C and B C do concurre in the point C,and ypon them faileth the line A B: therefore (by the conuerfe of the firftpeticionjthe two inward angles on one and the felfe fame fide,are leffe then two right angles.Wherefore the angles A B Cand CAB are leffe then two right angles : but the angles A B C and DBC arc (by the 1.3 proportion) equal to two fight angles . Wherefore the two angles A B C and 2? B C are greater then the two angles «A B Cand B AC. Wherfore taking away the angle zA B C,which i s com¬ mon to them both, there {hall, be left the angle B)B C greater then the angle B AC . And by tlie fame reafon, forafmuch as the two lines B A and CA concurre in the point A, and vppon them faileth the right line C2?,the two inward angles ABC and ACB are leffe then two right angles.But the angles zA B Cand B) B C are equall to two rightangles. Wherfore - the two angles ABC and D B C,are greater then the two angles AB C &.AC B. Wher¬ fore taking away the angle zA B C, which is common to them both, there fhal remaine the angle DBC greater then the angle ACB-. whichwasrequiredtobeproued. Here is to be noted,that when the fide of a triangle is drawen forth, the angle ofthe triangle which is nexttbe outward angle, is called an angle in order vnto it: and the other two angles of the triangle are called oppofite angles vnto it* UCotniUrj Of this Propofition followeth this Corrolkry , that ft is not poffible that from one & of this the felfe fame point fhould be drawen to one and the felfe fame right line, three equall frspojsuot*. right lines. For from one point, namely, A, if it be poffible,let there be drawen vnto the right line BB), thefe three equall right lines zA B,A C,8cA D. And forafmuch as A Bis equall to A C,the angles at the bafe are (by the fifth propofition Jequall. Wherfore the angle AB Cis equal to the angle ^ C.Z? .Agayne forafmuch as A Bis equall to A D, the angle ABB) is (by the fame) equall to the angle ADB : but the angle A B Cwas equall to the angle ACB. Where¬ fore the angle ACB is equall to the angle tAB)B : namely,the outward angle to the inwarde & oppo¬ site angle: whichisimpoffible.Wherforefrom one and the felfe fame point,can not be drawn to one & B the felfe fame tight line three equall right lynes: which was required to be proued. c A& other Cor - r&lLsrj follo¬ wing alfa of the ftme. By this propofition alfo may this be demonftratcd,that ifaright line felling \'pon two right lines, do make the outward angle equall to the inward and oppo- fice angle,thofe right lines fhallnot make a triangle, neither flial they concurre. For otherwife one St the felfe fame angle fliould be both greater,and alfo equal: which is impoffible*As for example. Suppofe that there be two right lines AB and C D, and vpon them let the right line B E fall, making the angles ABB) and CDE equall.Then I fay,that the right lines AB andCDfhallnotconcurre.Foriftheyconcurre,theforefaide angles abidyng equall, name!y,the angles C‘Z>£ and >2.5 D : Then forafmuch as the angle CDE is the out¬ ward angle it is of neceffitie greater then the inward and oppofite angle, &it is alfo e- qual vnto it : which is impoffible. Wherfore if the feid lines cocurre,the fhal not the an¬ gles remayne equalgbut the angle at the point B) ihall be encrcafed.For whether zA B abiding ofjzucmes memenm. FoLz6« abidirt^fixedyou fuppofe the line CD to be moued vnto itjfo chat they concurre, the fpace and diftaiice in the angle will be greater : for liow much more C X>approcheth to eABJo much farther of goethjt from D E. Or whether C D abiding fixed,you ima¬ gine the line tA B to be moued vnto it, fo that they concurre,the angle °X B ‘D will be leffe, for there¬ with all it commeth.ne re vnto the lines CD &BD, Or whether you imagine either of them to be mo¬ ued the one to the other, you ihall findc that the line sA B camming neere to C 23 , maketh the angled B E ldTe,and C D going farther from DE by reafon of his motion to the line B ^maketh the angle CDE to increafe. Wherefore it followeth of neceffitie,thatif it be a triang!e,and that the right lines *A B and C D do concnrre,the outward angle alfo {hall be greater then theinward and oppofite angle. For either the inward and oppofite angle abiding fixed,the outward is increafed: or the outwardea- biding fixed,the inward and oppofite is diminifhed: or els both of them being moued till they concurre,the inwarde is diminilhed, and the outwarde is more inereafed: And the caufe hereof is the motion of the right lines the one tending to that parte where it diminifheth the inwarde angle, the other tending to that part where it inereafeth the outward angle. B A C D E be io. Fheoreme. cTheiy9 Trogojition* <^n euery triangle gwo angle s> Vahich mo foemr be taken^ art lejie then Wo righ t angles. 2$ ^PP(sfe ‘that 'A B C be a gy 1 1 f triangle, Then I [aye that ^ two angles of the fay d tri* mm - - ■ i a Icjkihi mo-. right angles ! &xtend( by the Z-peticio) -A the line BC\ to the point '' v • * £ ~ ’ 'Q ID. And fora [much as (by the propoftion going before) the outward angle of the triangle A BC^ namely the amle A CD is greater then the inward and oppofite angle A B C: put the angle ACB comma, to them both therefore the angles AC D and AC B 'are greater then the an * gksABCand B C A .But (by: the 13 propoftwiftheangies A C -Dana AC B are equal! to mo right angles , VC ’here} ore the angles ABC and B CAare left then m/rigU angles . hi like fort alfo maygpe prone y that the angles B A Cand AC £ are left then mo right- angles l andaljo that the angles ,C AB <(3* ABC are left 'then. mo right angles. Wherefore in euery triqngftmo cm* gles Tthicb mo foeuer be taken ^re lejfe then mo right angiesmhsch Iras re* quindtoheproMd, "f ' V - . H.ii. ' Tins Gtnjtrfi8i$%a DemenftrutmS, ■A.. ThefirftHookp This may alfo be demonftrated without thchelpe ofthe former proportion, by the conuerfe ofthe fifth petition, and by the i$.propoficion as you faw was done in the former after Pclittrm. It may alfobe demonftrated without producing any of thefides ofthe tri* angle, after this maner. Another demt- ' SuPPofe that there a be triangle ABC And. in the fide B C take apointatalladuen- ftrxtion tnuen- turessand let the fame be D, and draw the line A D. And forafmuch as in thetriangle sedbj Proclus . A ^ D* the fide B D is produced, therefore ( by the former proposition ) the outward angle D C,is greater then the inward and oppofitc angle A B D. Agayne forafmuch as in the triangle A'D C, the fyde CD is produced, thereforefby the fame)the outward angle A D B,is greater then the inwarde and oppofitc angle D : but the angles at the point Darcequall to two right anglesfby the xj.propofition: jwherforc the angles A BCznd A CB are lefle then two right an¬ gles. And by the fame reafon may we proue that the an gles B AC and B C A are lefle then two right angles, if we take a poynt in the line <sA C, and draw a right line from it to the point B : and fo alfo may it be proued that the angles CtAB and *A BC are lefle the two ryght angles ,if there be taken in thelyne^^apoint, and from it be a line drawen to the point C. Corrolltry following this Progojitson, ConrtrHthon . -v By this proportion alfo maybe proued this Corrollary, that from oneand the feite fame point to one and the felfe fame right line, can not be drawen two perpendicular lines. For ifitbepoflible, from the point ^,Iet there be drawen vnto the right lrae2?C,two perpendicular lines AB^ and ACz wherefore the angles A B Cand CB are right angles. But forafmuch as A B Cisa triangle, therefore any two angles ther- of are (by this propofition ) lefle then two right angles.Where- fore the angles' Af J? C and A CB are lefle then two right angles: but they are alfo equall to two right angles.by reafon A B and <tA C arc perpendicular lines vpon B C: which is impofllble. Wherefore from one and the felfe fame point cannot be drawc to one and the felfe fame line two perpendicular lines j which was required to be proued. The ii. Theoreme . The i SJPropofition. In euery triangle , to the greater fide is fubtended the grea* Vppofe that ABC be a triangle , bauing the fide A C greater then the fide A B. Then l jay that the an* vie A'BC is greater then the angle B QA . For for* afmuch as A d is greater the A Bfput(by the 3 .pro - pojition) Vnto A B an equall line A & • And (by of buckoes blementes . roLij* the firfl petition) draw a tine from the point B to the point (D. And forajmnch as the outward angle of the triangle IDBC, namely } the angle aB>B isgrea* ter then the inittardand oppo/ite angle VCB (by the 16. propo/ition))but (by the )-. propofition) the angle AB)[ B is e quail to the angle A B B)Jor the Jyde A B is equal! to the jyde A ID; therefore the angle A BID is greater then the angle ACB. Wherefore the angle ABC is much greater then the angle A C B. Wherefore in euery triangle, to the greater Jyde isfubt ended the greater angle: trhich Snas required to be proued , You may alfo proue the angle at thepoint B greater then the angle at the point C (the fide A C being greater then ti elide AB Jif from the line ACyou cut of a linee- quall to the line sA 2?,beginning at the point C, as before you beganne at the point Ax and that after this manner. Let the line D C be equall to the line *A B and draw the line B D: and produce <iA B to the point E : A and put the line "B E equal to the line A D. Wherefore the whole line AEis equall to the whole line A C:draw aline from E to C, And forafmuch as AEis equal to A C, therfore the angle A E C is alfo equall to the angle A fE ("by the 5. propofition.’) but the angle ABCis greater then the angle cA E C.For one of the fides of the triangle CB E, namely, the fide B E is produced , and fo the outward angle <tA B C is greater then the inward and oppo- fite B EC (by the 1 6 propofition.-)wherefore the angle c A BC is & much greater then the angle &A C B r which was required to be proued. Notethat that which is here fpoken in this propofiti. bn* is to be vnderftanded in one and the felffame triangle. For itispoffiblcthat one and the felfe fame angle may be fubtended ofa greaterline,and ofa Idle line: and one and the felfe fame right line may fubtend a greater angle , and a ldfe an- gl e. As for example , Suppofe that there be an Ifofceles triangle AB C, & in the fide AB take the point D atall aduentures: & fro the line A C cut of(by the 5 .propofition) the lyne A E equall to the line AD. And draw a right line from D to .E.Wherfore the right lines DE and BC do fubtend the angle at the point A, & of them the one is greater, and the other lefle. And after the felfe fame manner a man may putinfinite right lines greater & lefle, fubtending the angle at the point A, Agayne fuppofe that ABC be an Ifofceles triangle. And let B C be lefle then either of the lines BA and AC . And vpon B C deferibe (by the firfl: Jan equilatcr trian¬ gle i? C D.And drawaline from A to 2); and produce it to the point E. And forafmuch as in the triangle A B A X>,the outward angle B D E,is greater rhen the inward & oppofxte angle BAD (by the i ^.propofition )And by the fame in the triagle ACD,tbe outward angle CDE,is greater then the inward &r oppofiteangleCAZ>:ther- fore the whole angle BDCis greater the the whole an¬ gle B AC .And one and the felfe fame right line fubten- deth both thefe angles,namelv,the greater angle & the lefle. And itis alio proued, thatgreatcr rightlines Sc lefle fnbtende one and the icIfefamcangle,Bntm K»uj* one Dcmonfratk#* An other da- menftrntnn af¬ ter Prophjrms* That which ti fpoken to thus Props fisen is re be Vnderftanded to one and the felfe fame tri¬ angle. Dtmwjlratsott leading to an impoffibiliti-e. This fropofition is A c Conner ft ibe former-. 9 An Affumpt is a. Propoftion tn kya of neceffitie to the helps of a demon f ration, the certainty te here of ss not Jo plabie, and ■thsrfore nedeth st felfefrfl to be d-cmossfirated. Art affumpt pu-t by Proclnsfor S he demonflra- tien of this Pro- psfstssn. \ cThejtrBcBoo%e <3neandthc fclfe fame triangle one right lincfubtendeth one angle. and the great zn^the great angleaand the lcITc the ldTe,as it was proued ThenfiTheoreme. The ipfPropoJilion* fn euery triangle , vnder the greater angle ujubtended the ‘greater fide* Appofe that A B Cbea triangle, hauyng t be angle A B C greater then the angle B C AT ben I fay that the fide AL is greater then y fide A BT 'or if riot. fife the fide AC is ether equal toy fide AByr els it is lejfie theitXbe fide A C is not equal to y fide ABfior then( by the $,pro* pofition )y angle ABC fhould beequall to the an * gle A C B: but (by fufipofitio) it is not, Where* fore the fide A C is not equal! to the fide A B. And the fide A C can not be lefie then the fide A3^ for then the angle A B Cfhoulde be lejfie then the angle A C B(by the prop ofitim nextgoyng before). But (by flip pofition it is not) Wherefore the fide AC is not lefie then the fide AB„ Wherefore the fide A C is greater then the fide A B. VAherefore in euery tri* angle r 'vndpr the greater angle is fnbtended the greater fide : 'tohich Taasreqm* red tg be demon fl rated. . - isi r 5 Da This propofition is the conuerfeof thepropofition next goingbeforc.VV,^ler-, fore as you fee.* that which was thecondufion in'theforroer,is m this the fuppo- iitionjor thing geuemand that which was there the thing geuen,is here the thing required or concluiion. And it is proued by ah argument leading to an impoffis biikieyis commonly all conucrfes are. ' Twins, demon jlrateth th is proportion after an other way : butfirftheputteth this ^AiTumpt following* -,n. , ;.q..y . :A1 If an angle of a triangle be deuide din to ttyo eefnallpartes, and if the HneXvhich dettideth it being dradren to the baff do deuide the fame into t\ho vneefuaH partes : the fides 'd’hich contaynt that angle jhalbe vneqttaH, and that fiialbe the greater fide, tyhicbfalleth on the grater fide of the bafc,andtha£ the lefie whtcbftHeth on the lefie fide of the baje. M ; - \ Cf y v . ' •• -xilJ ✓ill n: .... *•> rA. , ■ . i isVij Suppofe pA B C to be a triangle,and(by the p . proportion) deuide the angle at the point A,m\p two equall partes,by the rightfitfe A jD. And let the line ^tD deuide the bafe B Quito two vnequailpartes^ andle.t thbpart CD be greater, then the parteiMfc. Then I fay .that the fide AC is greater thenthe fide B. Produce the fine A D to the point Qand(by the third Jput-the line D i? ’equal! to the fine D A. And forafinuch as DC is by fuppoikion greater then DA, put (by the 3 .propofitionjD Fequal to ADS and draw a line fro E to Qand produeeitto thepoint G , Now forafinuch as .^Dise* quali to ED and D A is equall to D Ft therfore in the two triangles AB D.and EFD* two.fides of the one are .equal! to two fides of the other , eche to his correfp.ondenj fide’; and ( by the 1 5 ■. proportion) they contayae equall angle$5 nameiy3 the hed ari-- ofEucUdes Elementes. Fo!.i2* gies : wherfore f by the fourth propor¬ tion ) the bafe B A is equall to the bale £ F: and the angle D E F is equall to the angle D A B. But the angle DAG is by conftru&ion equall to the fame angle D zA B: wherefore (by the firft common fentence ) the angles EA(j and AE G are equall. Wherefore (by the 6. propo- fion ) the fide AG is equall to the fide E (7,Wherfore the fide A Cis greater then the fide £ G . Wherefore it is much grea¬ ter then the fide££.Butthe fide ££i$ equall to the fide A 3, as it hath bene proued. Wherefore the fide A Cis grea¬ ter then the fide A £j which was requi¬ red to be proued. This aifumptbeing put,this Proportion isofProclus thus dcmonftrated. Suppofe A 3 Cto be a triangle, hauing hisangle at the point £ greater then the an gle at the point C, Then I fay that the fide A Cis greater then the fide AB.Deuide the line B C into two equall partes in the point £>,and draw a line from A to£>. And pro¬ duce the line A D to the point £ : and put the line D £ equall to the line A Z>,and draw alinefrom£to£.Nowforafmuchas££> is equall to DC, and AD is equall to DE therefore in the two triangles A D C and B DE, two fides of the one are equall to two fides of the other, ech to his correfpondent fide, and they containe equall an¬ gles (by the i propofition). -wherefore (by the fourth proportion) the bafe££ is equall to the bafe AC, and the angle DB Eis equal to the angle at the point f^De- uide alfo th’angle ABE into two equal parts by the line B F: wherfore the line EE is greater then the line F A. And forafutuch as in the triangle A B £,the angle at the point B is deuided into two equall partes by the right line B F, and the line £ £ is greater then the line A F : therefore by the former AJfumpt the fide £ £ is greater then the fide BA: but the line £ £ is equall to the line A C. Wherfore the fydcAC is greater then the fide A B: which was required to be proued. The i i.lhenreme. The 20. Tropofition . In euery triangle tier 0 fides, which two fides fioeuer be taken, are greater then the fide remajning. A Vuppofe that ABC be a triangle. Then IJay that t^o fides of the triangle A B C, which two fides foe * uer be taken , are greater : 'then the fide remayning ^ j that is, the fides B A and ~ AC are greater then the Bdifi fit An ether de- monff ration afs ter Preelm. Gtmtfrucffc >#. &emsvj?r£ts$n. \A» ether dema ■* flrttoton with¬ out producing sits of the ftdcs . jht ether De- monfimtkn9 * TTjeJirH^oofy ftd c B C:and the (Ides A B and B C then the fide A C: and the fides A C and B C then the fide B AfProduce (by the 2,peticion)the line © A to the point D,And (by the third propo/ition finto the line A C put an equall line A 3); and dram a line from the point D tothepointe C. And forafi much as the line DA is equall to the line A Cohere* jore(by the ^,propoftion)the angle A3) C js equall to the angle A C D.But the angle BCD is greater then t he angle A C D,t here fore the angle BCD is greater then the angle ADC, And farafmnch as D CB is a triangle y bauing the angle BCD greater then the angle A D C,but(by the i$,propofitton)viu der the greater angle is fubtended the greater fide: Tt> herfore D B is greater then B C. But the lineD B is equall to the lines A Band A C(for the line JD is equall to the line A C) If herfore the files B A and A C, are greater then the fide BC.And in like forte may foe prone f hat the fides A Band B C are greater then the fide A C:& that t he fides B C and C A are greater then the fide A B VVherfore in euery t riant gle tlfo (idesftohich two tides focuer be taken, are greater then the fide remay * ning\ ifhich IVas required to be demonfir ated. This Proposition may alfobe detnonftrated withoutproducing any of the fides, alter this mancr. Suppofe ABCt o be triangle. Then I fay, that the two fides AB and A Care grea¬ ter then the fide B C: deuide the angle at the point A (by the 9. propofition)into two equal! partes by the right line <tA E. Andforafinuch as in the triangle^ B £,theoufr* ward angled EC is greater then the angle# ex/ E (by the i<5propofition),and the angle BAEis put to be equall to the angle E A (^therefore the fide AC is grea¬ ter then the fide (fE, And by the fame reafon the fide A Bis greater the the fide B £JFor in the triangle AEC the outward angle AEBtis greater then the angle CA <£\that is then the angle Wherelorealfb the fide AB is greater then the fide BE. Wherfore the fides A'B and A Care greater then the whole fide B C. And after the fame maner may you proue touching the other fides alfo. The fame may y etalfo be demonfi: rated an other way. Suppofe A B C to be a triangle. Now if A B C be an equilater triangle, then without doubt any two fides thereof are greater then the third. For the three fides being equallany two fides of them are double to the third.But if it be an Ifofceles triangle, either the bafe is lefie then either of the equall fides or it is greater. If the bafe be Iefle, then againe two of them arc greater then thethirde, butif the bafe be greater (let BC being the bafe of the Ifofceles triangle ABC be greater the either of the fides AB Sc AC and from it cut of (by the j.pro- ‘ pofition Polity* portion ) & line equall to any one of the other fides, whiche let bee f-E^and dtawe a line from A to E. And forafmuch as in the triangle A E B, the angle A E C is an out¬ ward angles therefore it is greater then the. angle B AE (by the 1 6. proportion) .And by the fame reafomthe angle AEB is greater then the angle CAE . Wherefore the an¬ gles at the poiht E ate greater theh the whole angle at the pointe A. But the angle B E A is equal to the angle B A E (by the f * propofition) for Ce B is put to be equall to B : £. Wherefore the angle remayning A Efts greater then the angle CAE, Whereforei’ alfo the fide A C is greater then the fide E (*. But the fide AB is equall to the fide 2? £* Wherefore the fides A B and Care greater then the fide B C, But if the triangle A B C be a Scalenum, let the fide A B be the greateft, and let A C be the meane, and B C the leaft. Wherefore the greateft fide being added tip a- ^ ny one of the two fides mull nedes be greater then the third. For ofit felfe it is greater then any of them . But ifAB being the greateft,you would proue the fides AC X / \\ and CB to be greater then it. Then as you did in the I- / / \\ » foceles triangle, cut of from the greateft a line equall to one of them, and from the point Cto the point of the interfedtion draw a right line,and reafon as you did be fore by the outward angles of the triangle,andyou ihal haueyourpurpofe. This propofition may yet moreouerbedemonftratedby an argument lea¬ ding to an abfurditie, and that after this manner. Suppofe ABC to be a triangle, then T fay that the fides A B and A C, are greater then the fide B C. For if they be not greater,they are either equall or leffe. Firft let them be equall,and from the line B C cut of the line B E equall to the line A B( by the ^propofition) wher- fore the refidue E Cis equal! to A C.Now forafmuch as A B is equall to BE they fubtend equall angles. Like- wife forafmuch as «xr Cis equall to CE they fubtend e- qual angles.Wherfore the angles which are at the point fare equall to the angle swhiche are at the pointed, which is impoftiblef by the 1 6. propofition). But now let the fides A B and AC beleffethen the fide B C ,and from the line B C cut of ( by the 3 .propofi- tion)the line B D equall to the line A B ,and likewife fr5 the fame line B C cut of the line CE equall to the line A C. And forafmuch as is equall to f O, the an<de B 2> ^ alfo is equall to the angle BAD (by the fifthpro- pofition).Againe forafmuch as zsf C is equall to CE therefore(bythe fame) the angle CEAis equall to the angle £ *A C . Wherefore thefe two angles B D A and C € A are equall to thefe two angles B*AD and £ ^4 C, Agayne forafmuch as the angle EDA is the OtitwardU angle of the triangle ADC, therefore it is greater thenB D £ C theanglef^C Foritisgreaterthentheangle‘Z)^Cfbvthe id.nronofmnn ) by the fit me reafon/orafimuch as CE^tU tSc the outwaS angle of ^ therefore it is greater then the angle* A D (for it is greater then the an S!/e ) Wherfore the ang es * DAandCE A are greater then fhe two angfej aHaeA .CBut theywerealfo proued equal! vnto them .• which is impoffible. Wherefore the M'.sAB and A Cure neither equall to the fide * C, nor leffe thenit but neater And foalfo nuy itbeprouedofthereft.- 7 ? out ^reate,. Ana A -l "i -i . ;!.t « edif cthef deffto'* ftrttticn leading SeAnAfurdiM ThefirflcBooh$ ^tiriefe demon- , morehriefely demonftrate this proportion by Carapanus f mtion bj the definition ofa right line, which as we hauebefore declared isthus: A right line is 'definition 9f« tbefomeft extenfm or Jrafygbt that is dr ftiay be from one point to another. SNhcdore. any one mght H»et fide p fa triangle, for that it is a right line drawenfrom fome one point to Come other one point, is of neceffitie lliorter then the other twofides drawen from and tpjthc famepointes*- , t •' • -/ •3K %Ll- cufiOJilftOfoai ■ V sr-- 7', > ; V..; rt OK. 1 a aaftthfttget Epicurus fuch as followed himderided this propofition, not counting it IfifcfJt ftllfht ^ortV to be added m the number of proportions of Geometry for the eafines » thereof, for that it is mam fell euen to thefenfe. But not all thinges manifeft to veafon *nd c-«- fenfe, are ftraight wayes mam fell; to reafon and vnderllanding- Tt pertay neth to Jerfttmiingt one that is a teacher of Sciences , by profe and demonftration to render a cer- tayneand vndoubted reafon, why itibappeareth to thefenfe: and in thatonely confiffeth fcience. / - s or S: •. •. 'oJitottrtgtfBrn • ... . j^oi Theu^ThcoreMe. TheilfPropofition* fffrom the aides of one of the fide s ofa triangle, he drawn to any point vpithin the fayde triangle two right lines* thofe right lines fo draWenfalbe kjle then the two other jides of f a - the triangle, but jhaUcont dine the greater angle. Vppofe that ABC be a triangletand fro the endes oftheJide'B C ^namely fro the point e s 3 and C y let [there be drawen Taithin y j triangle right lines 3 \D and CD to y point IX Then! fay , that the lines 3 3) and C ID are lejs’e then the ether jides of the triangle, namely, then the Jides 3y{ and AC:md that the angle fbhich they contayne, namely, 3 DC, is greater then the angle 3 AC. Extend ( by the ficond petition) the line 3D to the point Et And for af much as ( by the Zo. pro* pojition) in euery triangle the ttrofides are greater then the fide remayning% therefore the t'&o Jides of the triangle A3 E, namely, the Jides A 3 and A Ey are greater then the fide E 3. Tut the lineE C common to them both. Where < fore the lines 3 A and A C, are greater then the lines 3 E and E C.Againe for * afrmb as ( by the famejin the triangle C E Dyhe tlt>o Jides C E and E D, are greater then yfide D C, puty line D 3 common to them botht Tbherforey lines CE and E B^are greater then thelinesCDand D 3. 3ut it isproued that the Ones 3 A and AC^are greater then the lines 3 E and ECj/Vhere fore the lines BA and A C3are much greater then the lines 3 D and D C. Agayneforafmuch \ as A fcf* Fol^o* as (by the 1 6. proportion) in ettery triangle ^theoutward angle is greater then the inward and oppojite angle y therefore the outward angle of the triangle C IDE9 namely fBB Ctis greater then the angle CEB). VV her e fore alfo (by the fame) the outward angle of the triangle A B E* namely 9 the angle CEB is greater then the angle B AC. But it is prouedythat the angle BBC is greater then the angle C EB.VVherfore theangle BCD Cis much greater then the ana gle B J C .Where fore if from the endes of one of thefidesofa triangkybe dra* Wen to any point within the fay de triangle tW o right lines itho/e right lines fo drawn Jhalbelefte then the two other fedes of the triangle , but Jhallcontayne the greater angle : which was required to be demonflrated. .tone:" '< r.iuzn IV :! In this propofition is exprefledjthac the two right lines drawen within the triangle,haue their beginning at the extremes of the fide ofth: triangle* Tor fio the one extreme of the fide of the tr langl e,and fro ttifome o nepo i n t of the lame fide, may be drawen two right lines within the triangle, which fliail be longer the the two outwardlines: which i s wonderfull and feemeth firaunge, that two right lines drawen vpon a par te of a line,fiiould be greater then two right lines drawen vpon the whole lme.And agay ne it is poffible from the one extreme of the fide ofatriangle,andfrom fome one point ofthefamefide to drawe two right lynes within the triangle which fiiall contain^ an anglelefic then the angle contayned yndcr the two outward lines* 3s"7V*wH ' As touching the firfi: part* Suppofe A B C to be a re&an- gle triangle/whofe right angle let be at the point B. And in the fide B C take a point at al aduentures, which let be D: and draw a right line fro AtoD Wherfore the line AD is greater then the line AB fby the ip.propofitio) From the line A D cut of ( by the thirde) a lineequallto the line A B which let be D E. And deuide the line E Ain to two equall partes in the point F(by the io. propofition ) And draw aline fro F to C. Now -£r:;i s "i i r.*1 iaoi -j'i : in - forifmuch zsAFCisz triangle.thcrfore the lines A Fund F Care greater th? r i • „ j ^ormerProP°htionJ.-but«^?' F is equal ro F-Eiarherforethe rmktr f and F ©are grater then the line A C. Arntthe line © B is equall to the line!^ s" wh E fore the right lines F ©and FD are greater then the right lines j? B^T/r r‘ are drawen within the triangle ABC, the one from one extreme of Other from appint in the fame fide* C ; whiche was required to be proued *** A ’ *r\ rf v.1 •» l 4 J Av* v*r> * equall fydes, and from the lyne TJC cuttcofalin * eauall to thr itn. t thirde propofition) whiche le t bee BV: and drawc a line from to© ^ mid in the li* line Thefirft'Bbo\ i Sine tA Drake a point at al aducnturcs, which let be £, &drawalinefrom,Cto£. Nowfor- afrnuch as the line is equal to the lyne S 2>1therfore(bythefiftpropofition)the angle B AD is equall to the angle BD A. And for- afmuch as in the triangle E D Cthe angle ED Bis an outwarde angle,therefore (by the id. proportion) it is greater then the inw ard and oppofiteangie DEC, Wherefore the angle B AD is greater then the angle DEC, Wherfore the angle B AC is much greater then the an¬ gle D EC: andtheanglcB ACis contained of the outward right lines,5 y^an:d A C? and the an gl eD EC is contayned of the inward right * lines D E and E C: which was required to be proued. By meancs of this propesfi- tio n alio i s defer ibed that ky nd of triangles, which contayncth fourefides, A s for example, .this figu re A B C.For it is retained, ot towel- fides B A, A C,C BjSC EB.But ii; hath onely three a$-. gle s,onc a t. the p o in t B,an other. atthepointA.andthe third at the point C. VVhereforethis prefent figure ABC is a qua- drilatertriangle: whichofolde B philofophcrs hath eucr bene counted wonderful!, And here is to be noted, that there is difference bc- tweqe a three fided figure, and a three angled figure. For not euer> figure hauing three angles hath alfo onely three fides5as ir is platnc to fee in this figure*, Likewife it is not all on,a figure to haue lower fides, and fower angles, Forafoure fided figure may hauc onely threangles,as in the former figure: andafoure angled figure may hauc fine fides, as in this figure to lo wing, And fo of allothcc figures, - - ! -:Vr i'* ’-it+UiU ofrtion • Ofthreright lines, which are equall to tbre right lines gem* to makg a triangle . ! Butitbehoueth two ofthofe lines, which mo/oeuer be taken, to be greater then the third. For that in mery triangle mo fides, which mo fides foeuer be taken, are grea* * greater then the fide remaining* Vppofe that the three right lines geue be A,B,C:of which let tiro of them, Tvhkh ttro fbeuer be taken* be greater then the third y that is let the lines AjBAe greater then the line C, and the lines Afijhen the line B,and the lines B, C* then the line \A . It is required of OwmtyMw » Jkmh lines AfBfijomake a tmnglefTake a right line baiting an appointed ende on the fide £>, and being infinite on the fide E. Andfbythe $ , propopti* m)putVnto the line A, an e quail line : * ' * £) F,and put ImtQ the line Btan equall line FG^and'vntoy line C3an equall line GH- And making the centre Fyand the [pace b Ffiefcribe (by the $ .peticio) a circle D KJL. Agayne making the centre G0and the face G 77, deferibe (by the fame )a circle H I\L:and let the point of the inter [eHion of the fayd circles be K^andfby the fir fiptticion)dra'tr> a right line from the point fitoy point Fy Cst* an other from the point fito the point G, Then 7 fay 3t hat of thre right lines equall to the lines AfBfijs made a triangle I\FG. Forforafmuch as the point Dem»»ffr*t;o» F is the centre of the circled) therefore (by the definition) the lineF D is equall to the line F If, But the line A is equall to the line F D Wherfore ( by tbefrfl common fentence ) the line F Jfjs equall to the line A. Agayne for* afmuch as the point G,is the centre of the circle L IfjA^therefcre(by the fame definition) the line G Kjs equall to the line G BBut the line C is equall to the lineG FL therefore (by t he first common fentence) the line K^G is equallto the UneCfBut the line FG is byfiuppofition equal to the line B: therefore tbefe three right lines GFfi K»,and Kfifi 3are equall to tbefe three right lines AfBy C. Wherefore of three right lines , that is, JfJF,F G, and G 2^ , which are equallto the thre right lines geuen that is to A ' B.C,is made a triangle K FG: M^as required to be done. ? ~ V CmjimStm „ An other conftru£tion,and dcraonftration after Flnffatcs* Suppofe that the three right lines be Andyntq fome one of them3namely, t,berc9mJ to C,put an equall line D £,and(by the fecond propofitionjfrom the point A, draw the anf line £ C/ , equall to the lme^: and (by the fame) vnto the point D put the lineDHe- Irffiifrlt frt hne/,Al;d makLlnS the centre the point Ey & the fpace E <?,defcribe a cir- ^ cleFG : bice wife making the centre the point .D, andthe fpace D H deferibe an o- t er circle which circles let cutte the one the other in the point F. And draw - - I.ii;, thefe % t v - y<v thefe lines © F and E F. Then I faye thatf©££isa triangle defcribed of § right lines equal! to the right lines -^,£,C.For forafmuch as the line©// is equall to the right line ^4, the line ©£. (hall alfobe equall to the fame light line For that the lines © H and © £, are drawen fro the centre to the circumferenceJ.Likewife foraf- much as E ( 7 is equall to E F(by the 1 5« definition) and the right line B is equall to the fame right line £ (7:ther fore the right line £ £ is equall to the right line/? : but the right line © £, was put to be equall to the right line C.Wherfore of three right lines £ ©,© F, and ££,Which equa} to three right lines’ geuen, ts4,B,Cyis defcribed a triangle: which was required to be done. » n:,-, . 5 h tnflances in this Frebleme. Fir ft inflame. & r r <-\ r * * • V •ci \ \ i . : . ■ i • • * cv, 'A*'. '-vJi v : v ' • *n In this proposition theaduerfary paradueoturc mil caul Ilshat the circles' fhall not cut the one the other (which thingfeuF/f putteth them to do)Butnosr ifthey cutte not the one the other, either they touch the one the ocher,or they axe diftaunte the one from the other . Fiiffcif itbepoflibleiec them tqoche the one the other: as in the figure here put (the conftruction thereof anfwercthto the conftruaion viEuclide). And forafmuch as F is the cen¬ tre of the circle© if.therfore the ; line © £ is equal tothe line f’M And forafmuch as the point C,is . . th e ce ntte o f the circle HL, ther- fore theliiie fi G\ is equall to the j \ line G M. Wherefore thefe two / \ lines © F, and G H, are equall to ( one line , namely , to F G . But they were put to be greater then 9 it : for the lines © F, F (?, and G \ H, were put to be eqiidll to the \ lines uery two of which ' are fuppofed to be greater then the thirds ; wherefore they are both greater, and alio equall,. tuami infiance which is impoffible. Agayne if it be poifibledct the circles be di- ftant the one from the other, as are the circles DK&ndH ©.And forafmuch as Fisthe centre of the circle © K, therfore the line © £ is equal to the line F/V.And forafmuch as g is the centre of the circle L H, .therefore the line HG is equall to the line G Af: wherefore the whole line F G is greater then the two lines© £, and c j Hy (for the line £ (/,exee- deth the lines © £, and G H, by the line NM Jhut it was fupjpo« ofEuclides Elemntes* FoL]i. fed that the linesDF and if Care greater thm the Mne FG : as alfo it was fuppofed that the lines A and C7wcre greater then the line F(for the line D Fisput to be equal! to the line A, and the line F G to the line 2?, and theline H G to th,e line C. ) Wherefore they are both greater and alfo equall: which is impoffible. Wherefore the circles ney- ther tooch the one the other, nor are diftant the one from the other. Wherefore of ne- celfitie they cut the one the other: which was required to be proued. Ehe p. Tmbleme, The i^fPropofttwn. . v ' N\\ I s ' i Vpon a right line geuenyand to a point in it geuen: to make a reBiline angle equall to a reBiline angle geuen* Vppofey the right line ge¬ ne be AB, lety point in it geuen be A. And let alfo the reBiline angle gene be DCHJt is required Vpon the right line geuen A B,and to the point in it geuen A Jo make a reBiline angle e* quail to the reBiline angle geuen ID C H. Take in either of the lines CD and CHa point at all aduenturesjfr let the fame he Dand E.And(by the first peticion) dra'to a right line fro jp to E,And ofthre right lines yA F> F G and G jiJfbich let be equall to the three right lines geuen jhat is, to CDfD E,and EC,make(byy proportion. goyng hefore)a triangle, and let the fame he AEG; fo that let the line CD be e* quail to the line AF and the line C E to the line AG }and more oner the lyne D E to the line F G. And fora fmuch ds thefe t^o lines D C and CEare equall to Demonftr*tk*. thefe tV>o lines FA and A G,the one to the other, and the bafe D E is equall to the bafeF G'.ther fore (by the S. proportion) the angle D CE is equall to the angle F AG .W her fore vpon the right ImegeuenA B tand to the point in itge* uen namely A, is made a reBiline angle F A Gjqual to the reBiline angle geuen D CH: Tvhicb'SVas required to he done . . CiitftrH&lot?, An other conftrudionanddemonftration after Proclus, Suppofe that the right line geuen be A B: & let the point in it geuen be A, 8c let the Jn other rediline angle geue be C D E, It is required vpo the right line gene A B38c to the point jhuafLfZ de in it geue A,to make, a redijine angle equal to the rediline angle geue CD E. Drawe a m0n(l ration af s line fro C to E. And produce the line A B on either fide to the points F and G. And vn-, ter Proclm. I.iiii. to tt& er Tse- tswx-jtr# turn Pektariies, Tbefirmooke < t * s 4 — — — ■•• » * — — ***» .».*».* %* mw €q u al j yjj* j°r e- LlnC £.^PUC c^e ^ne 5(7 C(1 uallAnd making the cetre the point si, Sc the fpace AF, 4elcnbe a circle K F. And agayne making the centre the point B and the fpace B G de- fcribe an other circle G L: which dial of necefiitie cut the one the other,as we haue be¬ fore proued.Let them cut the one the other in the pointes M & N.And draw thefe right lines AN, AM, BN, and R M. And forafmuch as F A is equall to A M:and alfo to A N (by the definition of a circle)but C D is equall to F A,whcrfore the lines A M and A !M are eche equall to the line B C. Agayne forafmuch5 a sBG,is equall to B'M,and to B N, and B G is equall to C E: therfore either of thefe lines B M and B N is equal! to the line C E Bat the line BA is equall to the line D E.Wherfore thefe two lines B A & A M,arc equall to thefe two lines D E and D C.the one to the other.and the bafeB M is equal to thebafeC f.Wherforefby the 8,propofition)the angle M AB,is equall to the angle at the point D. And by the fame reafon the angle N A B.is equall to thefame angle at the. point O.Wherforevpon the right line geuen^i?,andtothe point in it geuen ^4,is de- feribed a reftilinc angle on either fide of the line ABi namely, on one fide the re&iline angle N A B,and on the other fide the re&iline angle M A B, either of which is equall to the re&iline angle geueh C BE: which was required to be done. An other conftru&ionalfo^and detnonftration after Pelitarius.’ Suppofe thar.the right line geuen be AB: and let the point in it geuenbe C, and letthere&ilineanglegeuen be DEFJtis required vpontheline geuen .//5,and to the point in it geuen -C,to deferibe a re&iiine angle equall to the reitiline angle geuen B E JF.Prqduce the line F E to the point G : and from the point E ere£fc(by the 1 1 .propofi- tion ) vnto the line GF a perpendictiler line E H, which, if it exa&ly agree with the lync iT'Z), then was the angle geue a right angle.Wherforeiffrom thepointe C you erefte a perpendiculer line vnto the line ABythzt fliall be done which was required to be done But if it do 1G K t> M notythenfrom the point//, ereft vn¬ to the line HE,& perpendiculer lyne H'B , whiche being produced Hull (by the fifth peticionjconcurre with the line £ D being alfo produced: for the angle B EH is Idfe then a right angle(whenas6’£//isarightangje), _ _ , _ _ , Wherefore let them concurre in the A C 3 G JS p point £>,audfois made the triangle JE> E H. After the fame maner fr5 the gsJica C,em% vnto the line A 3 a perpendiculer line C K ; which let be equal! to co .ini L the perpendicular line EH( by the % .pr6pofition):ahd frcini the point Ketcd vntothfi line K C>a perpendiculer line K Z^whiche let be equall to the perpendiculer lyne H D* And draw a line from C to E.Tben I fay thatirhs angle L C F,is equall to the angle ge¬ uen DE A. lor the two triangles HE CD and K G L,are (by the fourth propofition) e- qual, and cquilatcr the one to the other; and the two angles LCK a nd'DEH are equal* And the two angIes|5C/C and EEH are equal, for either of them is a right angle.Wher- fote ( by the i kComffion fentencejthe whole atigleZ# C i?,iseqtiall to the whole angle D E F. Which was required to be done. And if the perpendiculer line chaunce to fall without the angl egeuen4namely, if the angle geuen be an acute angle, the felfefame manner of demonftration will ferae: but onely that in ftede of the fecond common l'entencc, muft be vfed the 2,common fentencc* < • ■ ■ . ’ . ; ' ' ........ Appollonius putteth another conftru&ibn Si dcirionftratioh of this propolitio: which (though the demonftration thereof depende of proportions put in the third booke, y et for that the conftrufhon is very good for him that wil redcly , and mechanically, without demonftration, defenbe vpona line geuen, and toa point init geuen, a re&ilme angle equall to a re&ilinc angle geuen) I thought not amifle here to place it. And it is thus . Suppofe that the refliline angle geuen be CD E7 and let the right line geuen be ABS and let the point in it geuen be e^. Take in the line CU^a point at all ad¬ ventures , which let be.F. And making the cetre the point D, and the (pace D F, deferibea circle A (?,cut ting the line in the point G, and draw a ryght line from F to G , Likewife from the line AB, cut of a line equall to the linei> FjWhich let be AH. And Jq jj making the cetre the point ’ v A, and the fpace AH Ac- \ feribe a circle H K3 and from the point A/,fubtend vnto the circumference of the circle arightlyne equal! to the right line Fg , whicheletbee//FT; and drawe a right lyne from A to K. Then I fay thatthe angle HA KM equall to the angle CD E, The proofs whereofl now omitte for that it dependeth of the a 8 and 27 propofitions of the third booke. rdft etkiir tm- Jlraf~ion £<? menftrxtian *f~ ttT But now,as I fayd, by this you may very redily, and mechanically, without demon ftration, vpon a line geuen, and to a point init geuen deferibe a re&iline angle equall t6 to areftiline angle geuen. For ' in the rechlme angle geue, you neede onely to marke the two pointes, where the circumfe¬ rence of the circle cutteth the lines contayning the angle ge¬ uen: as the points Aand G: and likewife to marke in the line ge¬ uen asm AB,the point H, 8c fo a making the centre the point ^according to the fpace AH {which is put to be equal to FI)) deferibe a peece of a circumference on that fide that you wil haue the angle to he, asfor example the circumference H AT,and openingyour compaffe to the wideth from K*i» the ■®#Mpidei the firfi tnuenter of thii proportion. CtnUrtiftion, Dammft ration. the point F, to the point G, fet one foote thereof fixed iftthe point H, and marke the* point where the other foote cuttcth the fayde circumference , which point let be Kt And from that point to the point t^draw a right line: and fo (hall you hauedeferibed at the point A,zn angle equal tp the angle at the point I>. As in the figures in the end of the other fide put* 1 O empties was the firft inuenter of this proportion as witneffeth Eudemi us. 1 iSTbeoreme The Z^fPropofition. fftwo triangles haue two fide s of the one e quail to two fides of the other, ech to his correfpondent fide, and if the angle cotai- ned vnder the equall fides of the One , be greater then the an¬ gle contained vnder the equall fides of the other : the hafe alfo of the fame, Jhalbe greater then the baje of the other . V ppofe that there he Hbo triangles A B C, and 1) E F} hauing two. fides of the one , that is, A B, and ji C, e* quail to two fides of the other, that is jo IDE, and D F, ech to his correfpondent fideithat is, th fide JtB, to the | fide V E,and the fide A C to the fide DFx and fuppofe | ’that the angle $ AC he greater then the angle EVE 2 f hen I fay e that the ha/e BC, is greater then thebafe “ E Ft For forafmuch as the angle B AC is greater then the angle ED Fjnake (by the 23 -propofition)vp* on the right line DE,and to the point in itgeueDy an angle ED G equall to the angle geuen BAC. And to one of the fe lines, that is, either to A Cjr D Eyput an equall tine D G. And (by the firfi peticio) draft aright line from the point G^td the point E, and an other from the point F, to the point G. And forafmuch as the line A B is equall to the line V E, and the line AC to the line D Gythe one toy other, and the angle BACis (by conflructm) equall to . 7 the angle ED G, therefore (by the q.propofition) the hafe B C, is equal! to she hafe E G.Agayne for as much as the line D G is equall to the line D Father* ( hy the 5, propofition)the angle DGFy is equall to the angle D F Gf/Vhere* fore the angle DE G is greater then the angle E G EjFFherefore the angle E F G is much greater then the angle EGF And forafmuch as EFG is a trian # gle, hauing the angle E F G greater then the angle E G F, and (by the 18. pro* pojition) Vnder the greater angle is fubtended the greater fide , therefore the fide E G is greater then the fide E FjBut the fide E G is equall to the fide B C; therefore the fide B C is greater then the fide E F, If therefore ttvo triangles ■ *• 'i hmi 1 ofEmMei Elementeu haue mo fides of throne equal} to mo fides of the other , echeto his correfponfi dent fide, and if the angle contayned Vnder the e quail fides of the one ,he grea¬ ter then the angle contayned Vnder the equall fides of the other : the bafealfo of the fame jhalbe greater then the baje of the other : Which Was required to be proued . In this Theoreme may be three cafes. For the angle S JD <7, beingputequalltothe angled A C, and the line D C^being put equall to the line AC}ind a line beingdrawen from £ to G, the line E G flulleither fall aboue'the line G F , or vpon it,or vnder it. £ «- ^wdemonilrationferueth, when the line CZfaUethabbue the line GFt as we haue before manifeltly feene. But if it fall vpon it,as in this figure here piit. Then forafmuch as the two lines AB and^C,are equal to the two lines D £andD (/,thebne to the other, and they contayne equall angles by con- ftrudion: therefore ('by the 4. propofitionj the bafe B C, is equall to the bafe E G : but the bafe £ <7, is greater then the bafe£F : wherforealfo the bafeFCyis greater then the bafe £ F : whichwas required to be proued* But now let the line £ (j, fall vnder the line £ F, as in the figure here put. And forafmuch as thefe two lines A B , andACt are equall to thefe two lines D E and D (j, the one to the other, and they contayne equall angles, therefore (by the 4. pro- pofition)the bafe B Qis equal to the bafe £ G.And forafmuch as within the triangle DEG , the two linnes F and F E> are fet vpon the fide D E: ther- fore f by the 2 1 .propofition)the lines JD F and F £ areieife then the outward lines *2) G and G Ex but theline D G is equal to theline D F. Wherfore the line G E is greater then the line E E. But G E is e- quall to B C. Wherefore the line B Cis greater the theline £F,Which was required to be proued. 7 hree ejfcs i/i this prop aft sen . The firft cafe *. Second cafe. Third cafe. T> (7, which arc equal, vnto the points K and H : and draw a line from F to G: wherefore (by thefecond part of the fifth propofitidn)thc anglesXF G and F G ft, which are vnder the bafe F <7, are equall therefore the angle E FG is greater then the angle FG F.Wherfore (by the 1 8 propofitionj the fide£ <7 is greater then the fide £ JF; but the bafe B C is equal vnto the bafe E G: Where¬ fore the bafe B C, is greater then the bafe £ F : Which was required to be proued. produce the line s -Di7 and V > £ An other de- monjlratien of . the third caf. It may peraduenture feme, that Euchde fiiould here in this proportion haue proued, that not oncly thebafes ofthe triangles are vnequall, but alfo that the as teas of the fame are vnequallr for fo in the fourth'pro pofit ion3 after he had pro- Kii» usd mty EneliZe "hers prouith *fot :ht areas of -the triangles to :&<l SsKSqttttlt, 'lifter thiff, 2Jrtpajh/evJ*M ft all fade the campttrifsn of ■ triangles, w heft ijides being equalljkefr bit** Jes and angles •at the tcppe art ineipnall. E>ems»f ration leading to ah abfterdttji ued the bate to be equall, he protied alfo the areas to be equally But hereto may be answered, that in equall angles and bafes,and vnequall angles and hafes , the confideration is not like* For the angles and bafes being equall, the triangles al¬ fo fhall ofnecefliticbe equall, but the angles and bafes being vnequall, the areas fliall not ofneceflitie be equall* For the triangles may both be equall and vne- quail : and that may be the greater, whiche bathe the greater angle, and thegred* ter bafe, and it may alfo be the lelfe. And for that caufe Euclide made no mend# on ofthe comparifon ofthe triangles* Whereof this alfo moughtbeacaiife,fot that to thedemonftration thereof are required ccrtayne Proportions concer- ning parallel lines, which vc are not as yet come vnto* Howbeitafter the ^7, proportion of tins booke you fhalfind the comparifon of thearcas of triangles* 'which haue their fides equall, and their bafes andangles at the toppe vticquall* The 1 6.cTheoremv. T'heifJPropofitiOtt* If mo triangles haue mo fides ofthe otteequaU tttmo fades ofthe other , eche to bis correfpondent fide, and if the bafe of the one be greater then the bafe ofthe other: the angle alfo of the fame cotayned vnder the equall right lines ffktitt begrea^ ter then the angle ofthe other* p Vppofe that there he ftoo triangles, A fB,Cy land B) E FJiauing vm>o fides 0/ tb’onefbat Us, A B ft nd AC squall toffyo (ides ofthe 0* therjhat is to B) Ey and B) F, ecb to his correfpon* dent fide, namely, the fide AB to the fide !DF, and the fide A C to the fyde B) F. 'But let (he bafe B C be greater then the bafe E FfTbe 1 fay, that the an* git B AC isgrea ter then the an g le EB)F% For if nbt , then is 'it either equall Vn(6 it, or left then its But the angle BAG is not equall to the angle. B ID F: for if it There equall, tire bafe alfo B C Jhouldfbythe ^propofitton) be equal to the bafe E F: but by fuppofition it is not.Wherfott the angle B AC is note* quail to the angle EDF.Neither alfo is the angle fe AC [left then the angfiE £> F: for then fhould the bafe BCbe lefie the the bafe £ F( by the former pro* pofitioiifBut by fuppofition it is notyFherfaef angle BAGsmilefie anAe E B> FJnd it is already protied, that it isnoietiuafomitMerfdrefia^ Ae B AC is greater then the angle EB)F. If therfiret^o triangles haue iM fides of the one equall to nvo fides ofthe other, eche to his comfp*ndentfide,& if the bafe ofthe one be greater then the bafe ofthe other , the angle alfd ofthi fame contaytied Vnder y equal right lides jhdU be greater the the angle of theo . Tbit birstrs eafis i» this dttmssflnt- tiers . fir&i*fi.'t oj-tsmttm mtmmtes. Fol.tf. This propofitio a is plaine oppofiretb the el^ti is the coucrfc of the fdure Hr/comZoZh and twenty which went befor^.^nd it is proifed(as commonly ail eonuetfes are) inJsreaij <u- by a reafon leading to an ablufdi cie*But it may after Menelaas Alexand rintrs be monftr*ted- ^ demonilrared directly jafter this maner* ' fUlll! Mentlaus AUtr* andrin&s* Suppofe that there be two triangles iA B C 8c B £ F:haning the two fides A B and Ca C equal to the two Tides lD E and® £,the one to the other:andle£ the bafe B C be greater then the bafe E /.Then / fay that the angle at the point A, is greater the the aft^ gle at the point ®. For from the bafe BCe ut of (by the thi rde ja lineeqnallto the bale £ F, and let the fame be B (?.And vpon the line GB and to the point B putf by the 2 3,propofition)an angle equal to the angle © E Ft which let be G B H: and let the line B H be equall to the line D £.And drawe a lync from H to (band produce it beyond the point G: whiche being produced lhall fal either vppon the angle Ay or vpon the line A B, or vpon the line A C , Firffc' let it fall vpon theangle A. And forafmuch as thefe two lines B G and B H are equall to thefe two lines E F and E D} the one to the other,and they contayne e- quallanglesfbyconftru&ionjnamelyjtheangles G BH and D E F: therforef by the 4.propofition)thei bafe G H is equall to the D Ft and the angle® H G to the angle E D F.Agayne forafmuch as the line B His equall to the jline B A(( or the line A Bis fuppofed to be equal to the line D E, vnto which line the ineBH isput equal ) therforef by the 5 .propofition)the anglcBFIA is equall tothe angle BAH : wherfore alfo the angle E D F is equal to the angle B A //.But the angle B A C is greater then the angle BA Hi wherfore alio the an gle BAG is greater then the angleEDF. * ^ But now let it fall vpon the line in the point K, and drawe a line from A to H. And for¬ afmuch as thefe two lines B G andB H are equall to thefe two lines E F and E D3the one to theother,& they containe eq ual angles(by conftrudion j liamc- lv,the angles GBH andD E F: therfore (by the 4* propofition)the bafeGHis equall to the bafeD F# and the angle B H G to the angle EDF.Agayne for¬ afmuch as in the triangle B A H,the fide BA is equal to the fideB H,therfore (by the 5 .propofition)the angle B A FI is equal to the angle B H A.But thean¬ gle B H A is greater then the angle B H G; wherfore alfo th e angle BAH is greater then the angle BHG, Wherfore the angle B A C is much greater then the angle B H G.But it is proued that the angle BHG is equall to the angle at the point D . Wherefore the dngleE.AC is greater then the angle at thepbinte £): Which was required to be proued* But now fuppofe that the line H G beyng produced doo fall vppon the line zAC, namely,in the point K.And agayne draw alfo a line from cA to //.And forafmuch as B G i s equal! to E F,and B H,to E /^therefore thefe two lines B G and BH are equall to thefe two lines E F and E D ,the one to the other,and (by conftrudion Jthey contayne equall angles,namely,the angles GBH and FED, Wherfore (by the fourth propofitio ) K.iii. the Thirdcstfe, je» ttfter de- ■xns nfiratisn /tf‘ Ter Hers Mea eebaaicm. chc bafejG His equal tothebafc *Z) Ft Sc th’an gle B H G is cquall to th’angle E D E, And for¬ afmuch as GH is equall toDi^andDF ist- quall to AC: therforeG H alfo is equal! to A C. Wherfore H K is greater then A C, where¬ fore H K is much greater then A K. Wherfore (bytheiS. propofition) theangle KAH is greater then the angle K H A. Agayne foraf- much a sB H is equahto A Bl( for B H is pute- quall to D £, which is by fuppofition equal to B the r fore (by the 5 .propofition ^ thean¬ gle B H A is equall to the angle B AH. Wher- fore the whole angle BHK is leflethen the whole angle BAK, But it hath bene proued, that the angle B H K is equall to the angle at •the point D,wherfore the angle B AC is grea¬ ter-then tlie angle at the point D, which was -re q u ir ed to b e p r o ued. Hero Mcchanicus alfo dcmonftratethican other way,andthat by 4 dixcft •demonftranon. -a Suppofe that there be two triangles ABC and DEE, hauyngthetwofides-42?,and.// C,equallto the twofidesD E,8cDF, tlveone to the .other, and letthe bale BCy be greater then the bale E E. Then I fay,that the angle at the point -4, is greater then the angle at the point -D.Forforafmuch a sBCt isgreater the £F,produce£Ftothepoint£7, and put the line E G, equall t“o the line 2 EC. likewife pro¬ duce the line D E to the point //, and put the line D Hy equall to the line D E,. Wherefore making the centre the point D,and the /pace D F,dcfcribe a circle,and it fliall pa'fie alfo by the paint //.Let the fame circle be E K //.And forafmuch as A C and A B are greater the BC (by the 2 0.propofitio)& the lines AB & A C, ar equal to the line £H,8c the line BC is equal to the line E (/.Therefore theline E H is grea ter then theline E (J, VVherefore making the centre the point £ and the fpace E G deferibe 3tcirde,and it (hall cut the line E //. Let the fame circle be G K: and from the common ife&ib of the circles, which let be the’point if, draw rhefe right lines K D and If E. And for¬ afmuch as the point D is the centre of the cir¬ cle H K F, therefore^ by the 1 5, definition JltheliijeD K,is equall to the line D H, that is vnto the fine A C. Agayne forafmuch as E is the centre of the circle G K, therefore •theline E If is equal to theline EG,thatis,to the line B C. And forafmuch as thefetwo lines A B and A C,are equall to thefe two lines D E and D A", and the bale 2? C is equal to the bale E K(t or E Kis equall to E G (by the 1 5 .definition) & EG is put to be equal to B C).Whereforef by the 4.propofitidn)the angle B A Cis equal to the angle E D K, But the angle E D Ki s greater then the angle ED F : wherefore alfo the angle B A C,if greater then the angle £ D Ftwhich was required to be proued. Tk 1 ofEuclides Elemenfes. FoL]6 . The 17. T htoreme. (Thei6fPropoftim . .#2 j _r j; “ ■' ’• * '' 1 - . •. s ; ' • • ffmo triangles haue two angles of the one equall to two an¬ gles of the other, ecb to his correftondent angle, and haue alfo one fide of the one equall to one fide of the other , either that fide which lie th betwene the equal! angles, or that which is fukended vnder one of the equallangles: the other jides alfo of the onefalbe equal! to the other fides of the other, eche to Us correfpondent fide, and the other angle of the one jhalbe equal! to the other angle of the other. Vppofe that there he t*too triangles A B C,atid D E E,hauing Vtoo angles of the one , that is .the angles A BC,andB C A, e quail to n>o angles of the other, that is, to the angles ID E F,and E FD,ech to his correfpo dent angle, that is, the angle AB C, to the angle D EF,and the angle B C A to the angle EFDyind one fide of the one e quail to one fide of $ other, firft that fide Ttshich lieth betlpene the equall angles , that is, the fide BC, to the fide EF.The I fay that the other fides alfo of the one fhalbe equall to the other fides of the other, ecb to his cor re* f pendent fide,thatis,the fide A B,to the fide V E,and the fide A C,to the fide DF,and the other angle of the one, to the other angle of the other, that is, the angle BA C to the angle E ID FJFor if the fide A B be not equall to the fide T> Ejthe one of them is greater , Let the fyde A B be greater : and ( by the 5 .propo fiiion) Vnto the line T> E,put an equall line GB,and drafts a right line from the point G,to the point C. ISLoSts for a [much as thedine G B, is equall to the line D E,ahd the line BC to the line E F ^therefore thefe tstso lines G B and B C, are equall to thefe tSt>o lines ID E and EFjhe one to the other, and the angle GBC is (by fuppofition) equall to the angle D E F.V therefore (by the jp.propojy* tion ) the bafe G Cis equall to the bafe T> F, and the triangle GCB is equall to the triangle DUE, and the angles remayningare equall to the angles remay • ning vnderTvhichare fubtended equall fydes .Wherefore the angle GCB is o quail to the angle DFE.But the angle D EE is fuppofed to be equall to the an gle BC A.W here for e(by the firfi common jentence)the angle BCG'ts equal to the angle B CA,the lefie angle to the greater -.fvbich is impofiible . Where- fore the line A B is not Vne quail to the line 'DE.Wherefore it is equall And the the line B C is equall to the line E F:now therefore there are tUso fydes A B and Kjiq. BC Dentonfirati.n leading to an abfurditk* iB C equal! to tV>o fydes t> E and S fi , the one to tbeother,and the angle AftC, if equal! to the angleD EE .Wherefore(by the 4-propofition)the bafe AC is equal! to the bafe D E^and the angle remay ning ft AC is equal! to the angle re mayning ED F, Agaynefuppofe that the fydes ful tending the equaU angles be equall the one to the other Jet the fyde I fay AD be equal! to the fyde D E. Then agayne l fiyjkat the other fydes of the one are equal! to the other fydes o f the other ,ech to his correfpondent fyde, that is the fyde AC to the fyde D E,and the fyde ft C to the fyde E F : and moreouer the angle remayning, namely , ft A C, is equall to the angle remayningy that is, to the angle E VF. For if the fyde ft C he not equall to the fyde E F,the one of them is greater: let the fyde ftCJfit be pofii* He , be greater . And (by the third proportion) Vnto the line EF,put an equall line ft H,and dr an? e a right line from the point J A to the point 0, And forafmuch as the line ft Bis equall to the lineE E , and the line A ft to the line D E, therefore thefe tn?oJydes A ft and ft 0, are equall to thefe tn>o fydes D Eand EF, the one to the other, and they contdine equa 11 angles, V Vi here fore (by the A.propofitmi) the bafe A His equall to the bafe D E,and the triangle A ft H,is equall to the triangle D E F^and the an* c gtes remayning are equall todhe angles remayning, Vnder'tobtcb ar fubteded equalfydes.VVherfore the angle ft HA is equall to the angle FED. ftut the angle E ED is equall to the angle ftCJ \ Where * fore the angle ft HA is equal to the angle ft C A \ Wherefore the outward angle of y triangle A HC, namely, the angle ft 0 Ads equall to the inlvard and oppofite angle yiamely , to the angle HCAft»hich(by the \6 proportion) is impofible.Wherfore the fyde E Fis not' Unequal! to the fyde ft Cohere fore it is equall. And the fyde A ft is equall toy fyde D Ei'toherefor e thefe t'Hoo fydes Aft andftC,are equall to thefe two fydes DE and.E F,the one to the other, and they contayne equall angles : W her fore (by the ^propofitm )the bafe A C is equall to the bafe D E:and the triangle A ft C,is equall to the triangle D EF,and the angle remay ning,name* fy,the angle ft AC is equall to the angle remayning, that is, to the angle EDF. If there fore two triangles haue Wo angles of the one equall to tH>o angles of the other, ech to his correfpondent angle, and haue alfo one fyde of the one equall to qne. fyde of the other, either thatfydefohich lietb betToene the equall angles, or that tohich is fub tended Vnder one of the equall angles : the other fydes alfo of the one fhalbe equall to the other fydes of the other, eche to his correfpondent fide, and the other angle of the one fhalbe equall to the other angle of the other : Ifibicb laces required to be proued* Whereas in this propofition it is fay de, that triangles are equall , which featuring two angles of the one equall to two angles ofthe other, the one to the o- thcr, of Euclides Elements . FoLff. ther, haue alfo one fide ofthc one cquall to one fide of the other,either that fide which lieth betwene the equal! angles, or that fide which fubtendeth one of the equall angles :th is is to be noted that without that caution touching the cquall fiie,the propofition fiiall not alway e$ be true. As for example. Suppofe that there be a rectangle triangle A B C.whofe right angle let be at the point B,Sc let the fide B C be greater the the fide B ^4:and produce the line A 2?,f ro the point 'B to the point D.And vpo the right line B C & to the point in it C, make vnto the angle B AC an equal angle(by the 2 3 . propofition), which let be BCD ,8c let the lines BD Sc CD, be ing produced cocurrein the point D .Now thS there are two triangles A B C,and BCD,w hich haue two angles of the one equall to two an¬ gles of the other,the one to the other,namely, the angle *ABC to the angle DBC (for they are both right angles ), Sc the angle B A C, to the angle BCD( by conftru&ionjand haue al¬ fo one fide of the one equall to one fide of the other, namely, the fide B C, which is co¬ ition to them both. And yet notwithftanding the triangles are not equall :for the tri¬ angle B DC,is greater then the triangle AB C.Forvpon the right line BC, and to the point in it C,defcribe an angle equall to the angled CB: which let be FCB( by the 2 3 . propofition ).And forafmuch as the fide B C was fuppofed to be greater then the fide, AB , therefore (by the 1 S.propofition) the angle B AC is greater then the angle BC ^,whereforealfotheangle5CDisgreaterthentheangle^CJP. Wherefore the tri¬ angle BCDis greater then the triangle B £f. Agayne forafmuch as there are two tri¬ angles A B £and B Cishauing two angles of the one equal to two angles of the other*, the one to the other,namely,the angle ABC to the angle FBCffor they are both right angles) and the angled CB to the angle FCB(by conftru&ion),and one fide of the one is equall to one fide of the other, namely,that fide which lieth betwene the equall an- gles.thatis.the fide B C which is common to both triangles. Wherefore (by this pro¬ pofition) the triangles A B Cand F B Care equal. But the triangle DBC is greater the the triangle F2?C. Wherefore alfo the triangle D B fis greater then the triangle A B C. Wherefore the triangles ABC and D B C, are not cquall.-notwithftanding they haue two angles of the one equall to two angles of the other,the one to the other , and one fide of the one equall to one fide of the other. The reafon wherof is, for that the equal fide in one triangle, fubtedeth one of the equall angles, and in the other lieth betwene the equal angles. So that you fee that it is ol necclfitie that the equall fide do in both triangles, cither fubtend one of the equall anglcs,or lie betwene the equall angles. Of this propofition was Thales Milefius theinuentor, as witncffethEude- inus in his booke of Geometricall enarrations, Thales the snuentcr of this fro fojstitss. clhe i&FTheoreme. Theij-Tropo/ttion* If a right line falling vpon two right lines Jo make the alter ~ nate angles equall the one to the other : thofe two right lines are parallels the one to the other . c . ThefrflTdooke \ V ppofe that the right line E F falling Vppon thefe Mo right lines A B \and C D^o wake the alternate angles ^namely, the angles A EF<& E *F V equall the one to the other Ah enl fay that A B is a parallel line to C D. For if not 3 then thefe lines produced fhall mete together ^either on the fide of B and (Dyor on 23 njlrattan the fyde of A ip C.Let them be produced therfore , let ri)em mete lflt be pofiible on the fyde of B andVjn the point G. VF her fore in the triangle G E Ffthe ouMar dangle A E Fis equal to the m - trard and oppofite angle E F Gftohich (by the f6- propo(ition}isimpofiible4 Wberfore the lines AB and C(D beyng produced on the fide ofB and jhallnot meeteJn like forte alfo may it be prouedthat they fhall not mete on the fyde of AandC , But lines Tbhiche being produced on no fydemeete together yare parrallell lines (by the lafi definition:)wherforeA B is a parrallel line toC T)Jf therfore a right line fallings pon Mo right lines ,do make the alternate angles equall the one to the other: thofe Mo right lines are parrallels the one to the o* then Ttthkh 'to&s required to be demonstrated . Tbit mrde al¬ ternate 9 ft A in tkmcn fenfet*. Hots it is hdgtt in this plats. Whu h singles are csUedalters fusts. This worde alternate is ofEuclide in diuers places diuerfiy taken: fomtimes forakind offituation in place,andfomtimefbran order in proportion, in which fignification he vfeth it in the v.booke,and in his bokes of numbers. And in the firftfignification he vfeth it here in this place, and generally in all hys other bokes ,hauing to do with lines Sc figures* And thofe two angles hecallech alter¬ nate, which beyng both contayned within two parallel or equidiftant lynes arc neither angles in order, nor are on the one and felfe fame fide, but are feperated the one from the other by the line which falleth on the two lines: the one angle beyngaboue,and the other beneath. T he iy.Theoreme. The z8. Tropoftion. ffa right line fatting ypon two right lines ,ma^e the outward angle equall to the inward and oppofite angle on one and the fame fyde, or the inwar de angles on one and the fame fyde , e- quail to two right angles ithoje two right lines ) hall be paraU lets the one to the other • Vppofethat the right line EF^ fallyng Vppon thefe Mo right lines A B andCfDfio make the outward angle EGB equall to the inward and oppofite angle G H T>,or do make the inward angles on one and the of Euelides Element es. FoLfi. the fme jidejbat is, the angles BGH andGHD equall to tibo right anglesXhen I fay that the lyne A IB is a parallel line to the lyne C DEorforafmuchas the angle E G Bis(byfuppofition)equall to. the an* gle G HD^and the angle E G B is(by the 1 5 .pro* pofition)equdll to the angle jiGH: therfore the an* gle AJj... ft he quail to the angle G HD : and they are alternate angles.VVherj or e( by the zy. pro pop* tUn) AB is a parallel line to C ID. Agaynefdrafmuch as the angles BGH and G HD are (by fuppo/ition)e» quail ionvo right angles & ( by the i^propofition )tbe angles J. G H and BG H,ar ealfo equall to tVo right angles, therefore the angles AG Hand'B G H} are e quail to the angles BGH and G HD: take avay the angle BGH vhich is common to them both Wherfore the angle nmainyng,namety,AGH is equall to the angle remay ning,namely, to G H D.And they are alternate an* gles. VVherfore(by the former proportion) A B isaparallell line to CD , If therfore a right line fallyng Vpon tvo right lines, do make the outward angle e* quail to the inward and opposite angle on one and the fame fide, or the invar de angles on one and the fame (ideyequall to tVo right angles, thofe tVo right lines Jhall be parallels the one to the otherivhich Vas required to be proued , Ptolomeus dcmonftrateth the fecond part of this propofition, namely, that the two inward angles on one and the fame fide being cqual^the right lines arc parellels,afterthis manner* Suppofe that there be two right lines A B and C T), and let a certaync right line E FCj H cuttethemin fuche forte, that it make the angles B F g and FGD e- quail to two right angles. Then I fay, that thele right lines zAB and CD are parallel lines,that is, they fhall not con- curre.For if it be poffibledet the lines B F and G D being produced concurrc in thepointeK. Noweforafmucheas the right line E F itandeth vppon the right line A 2?,therfore(by the 1 ^.proporti¬ on Jit maketh the angles zA F £, and B FE equall to t wo right angles : Iikewife forafmuch as the line E g ftandeth vpo the line C Z^therforef by the fame propofition Jit maketh the angles CG F and D g F equall to two right angles. Wherfore the foure angles B FE, zAF E, CG F, and ‘Z> C7 Fare equal to foure right angles : of which the two angles B F G and FGD are(by fuppofition) e- quall to two right angles,wherfore the angles remaining,namely, zAV G and C(?F are alfo equall to two right angles .if therfore the right lines F B and G D being produced (the inward angles being equall to two right angles Jdo concurre, then fhall the lynes FA3.n6.GC being produced concurre. For the angles AY G and CG F are equall to two right angles.For either the right linesfhall concurre on either fide, or els on nei¬ ther fide.For that on either fide the angles are equall to two right angles. Wherefore let the right lines FA and GC concurre in the point L.Wherefore the two right lines LAFKandLCGK do comprehends ipace, which (by the 6. petition) is impofiu L.ij, ble. S DmtnftrnMis An $ther cUm$- Jlrnthn of the fccondptsrt af this prtfffitien after i'tekmex* ThefirftBooke He.Wherfore itis not poflible that the inward angles being equal to two right angled the right lines fhould concurrc. Wherefore they are parallels : which was required to beproucd. ' c ^ 5 • n ittmznfifAUon Seeding ft <f* fjrftgart. The lo.Theoreme. The zyfPropofition. ... ' ■ ■ fright line line falling rppon two parallel right lines : ma- hpththe alternate angles equall the one to the other : andal- fo the oumarde angle equdtl to the inwarde andoppojite an- glean one and thejamefideiand moreouerthe inwarde an¬ gles on one and the fame fide equall to two right angles. Vppofe tlxttppon theft parallel lines A Band C IDd&fal the right line E F. Then I fay that the alternate an* gles ibbkk it maketh, namely r the angles A G H and G H Bf^are equad the one to the other. andy the out* li?ard angle EGBis equal \to the in'toarde and oppo* fite angle on the fame: fide 9 namely yto) angle G H B:andj the inward an¬ gles on one and the felfe fame fide ythat is^the an - c ales BG Band G HT>, are equad to two right Angles. For if the angle AGHbe not equal to the angle G H ID, the one of them is greater. Let the angle AGH he greater. And for af much as the angle AG His greater then the angle GH Vrput the angle B G Hcommo to theboth.Wherforey angles A GHand BGH, aregreater the y angles BG H&GB iD. But by ey i$ . propofitio) angles AGHzsrB GHare e quad to ttfo right angles, "toher fore y angles BGH er GHD arelefie the two right angles. But (hy) $. petition ) ifvpotWo right lines do fall a right line yaking) inward angles on one and y fame fide, le fie the Ctoo right angles, thofe right lines being inftmtly produced muft ncedesaty length meete on the. fide Ipherin are the angles kffe the tSto right angks^VVherfore the right lines A B and C ID being infinitely produced 'toillat) length meete. But they cannot meete, becau ft they are paradels(hyfiuppofition):^berfore the angle A G Bit not -vnequaU to the angle GHDi'tokerfore it is equad. And the angle A G His(by the i<;.propofition)equall to the angle EG % VVherfore (by the fir ft common fentence) the angle E ’Q Bis equad to the an* ^ But the angle B G H common to them botkwherfore the angles EG B and B GH,are equall to the angles BG Hand G HV.But the angles RGB and ofEucluIes Elements* FoL]p. md <BGH are (by the i^propofition)eqmM to ftoo ri*h t angles , VJ? he re fore the angles BGB and G HD are alfe equattmtm- right angles. If a lyne therfore do fall Vpo two parallel right lines:kmd^tly th£naltermte^arigles equal the one to the other.and alfo the outward itngieftq-udltd the inipard and oppoo fhe angle' on one and the fame fide\ and momm et the inward angles on one and the fame fyde equallto tSho right angles: yahiche 'Was required to he demon* Jirated, . This proportion is the conuerfe ofthe 'two proportions next going before. Fo^that which in cither of thernjs the thing fought,or c5cluron}is in this the tin ng geucnjor fupportiotfi And contrati wile the thiri'gcs which in them were geuen orfupportions3areinthis proued,ahdafecidhciiirons4 .4 ■. ' '• :v\ ‘ ''A f\ v V.'Vv V> : ’Telit arins after this proportion addeth this witty conchifioiv* ■ . . ; ' ' . . . - ... I- ' F ■ V ' •’ If two right lines tybich cut. Wo parallellipes filo be Went thefajde parallel lines conairrs in a point fdndmahe the alternate angles equally or the onward angle equall to the irWardand oppofite angle on the fame fide ,cr finally the tveo inward angles on one and the felfe fame fide, equall to Wo right angles uhofe Wo right Imes are : draft endiretlly and make one right line. Suppofe that there be two right lines ex/ B and Gif, which let cut two parallel lines JD £ and F G: and let ^ B cut the line D E in the pOintAAand let C B cut the line F G in the point K:8c let the lines A B ScC B,c oncurre betwene the two parallel lines DE@z F G in the point B : and let the angle D HBbt s- qualto the angle B K G : or let the angle A HD be equall to the angle B^K-F: orfinally let the angles B H D and B KF be equal to two right angles.Thc I fay that the two lines ABandBC are drawen di re£lly,.and do make one right line. For if they be not,then produce AB vnti! it cut FG in thepoint Ii.anH let A L he one righ t line, and fo ihal be made the triangle B L K. Now then ( by the firft part of this 29.pfopofition)the angle DHB dial be equal to the alternate angle (] L B: but(by fuppofitionj the angle D H Bis equall to the angle A KG. Wherefore the angle B L G is equall to the angle B KL} namely, the outward angle to the inwarde and oppofite angle:which (by the i <5.propofition Jis impolfible. Moreouer ( by the fecodpart of this 29. propofitio Jthe angle A-H-D lhalbe equal to the angle B L AT, namely, the outward angle to the inward and oppofite angle on one and the fame fide. But the fame angle AH D is fuppofed to be equal! to the angle B K Fiwherefore the angle B K Fis equall to theangle B L A'. Which (by the felfe fame 1 6. propofition) is impolfible. * , - . ( . . Laftly forafmuchas the angles B HT) and B KF are fuppofed to beeqtiall to twfO right angles, & the angles B H D &BLKatealfo by thelaftpart ofthis appropofiti- on equal to two right angles,therefore the angle B K F lhalbe equal to the angle-B LK; which agayne by the felfe fame i£-propofition is impolfible, IhezifEheoreme The ^.Tropofition* "Might lines which are parallels to one and the felfe fame right lineiare alfo parrallel lines the one to the other . L.Hj. Suppofe 7 his prop oft toss is the conuerfe ofthe two for¬ mer propofttisi. An addition of Pelitarim. Demonftratim. . leading tp an alfitrditie t Fir ft part. Second pare. Third part? ThefirftBooke Vppofetbat thefe right lines f, A B and CD, be parallel lines to theright line EF. Then I Jay, that the line A B is a parallel line foC<D.Let there fatlDpon the/e the lines a right line G HKf Andforaf • much as the right line G HI f falleth yppon thefe parallel right lines AB and EF,therfore(by the prop oft ion ~^VJ:uvV<\a going before) the angle A G H is e* quail to the angle G HR. Agayne for* ajmuch as the right line G If falleth yppon thefe parallel! right lines E F and C D ^therefore (by the fame) the A .. . a/ B E /h jr T. •> . -* i / r-£ -i >• ", * C ■ - - . ' ] f'r‘ ■ /& A •1; 1 T~ ' • • Anciheretefem- sh&e Prsile/zutf 1 angle GHF'u equal! to the angle G JfJD.lS{pTi> thenit proued that the angle A G H is equal! toy angle G HE, andy the angle G KJD is e quail to the angle G HF.V E her fore the angle jiG Kjs e quail to the angle G KJD, And they are alternate anglesiwherfore AB is a parallelline to C DRight lines therfore ft hid? are parallels to one and the felfefame right line, are alJo paraM lines the me to the other: Trbicb y?as required to be proued . EuciideirathedemonfkationofthisprQpofmQn, fetteth the two parallel Isneswhich are compared to one, in the extremes , and theparrallcl towhomc they are compared,he piacetli in the middle , for the eaficr demonftration . Ic may alfo be proued euen by a principle onely. For if they fhouldc conairrc oa any oncfide,thcy {houldconcurrealfo with the middle iinc^andfojfliould they not be parallels vnto it, which yet they are fuppofed to be* But if you will altertheir pofition and placing.andfet that line to which you will copare the other two lines ,aboue,or beneath: you may vie the famedemon fixation which you had before. As for example. Suppofe that the fines A B and CD be parallels to the fine £ F : and let both the lines A B and CD , be abouc , and let the line EF be beneath,and not in the middeft. Vpon which -let the right fine GH Ki all. And forafmuch as either of the angles KH DandAfC-Sisequall to the angle H KE9 (for they ar alternate angles Jtherforc they are (by the firft common feutence Jcquall 'the one to the other.Whereforefby the 28 propofition) the right lines AB and £F» are parallels. But here ifa man will obieft that the lines E K and K F, are parallels vnto she line C D,and therefore are parallels the one to theother. V Ve will anfwere that die lines E K and K Fare partes of one parallel line,and are not two parallel lines. A A B c V E A r FoL\ o. liaes.For parallel lines a r vnderffandedtobe produced infirmly But E K being producedifalleth vpou K F*Wherefore it is one and the felfe fame with it, and not an other, wherefore all the partes of a parrallelline are parallel s , both to the right line vnto which the whole parallel line is a paralleled alfo to al the parts of the fame right line* As thelineEK is a parallel vnto HD,and the lineK Eto the line C H* For if they be produced infinitly ,they will neuer conciirre, Howbeit there are fome which like not, thattwo diftindt parellel lines, fhouldbe taken and counted for one parallel line: for that the continuall quan¬ tity ,namely, the line is cutafonder,andcefTeth to be one* Wherefore they fay, that there ought to be two diftind parallel lines compared to one. And therfore they adde to the proportion acorredion, in this man er. Two lines being parallels to one line ; are either parallels the one the other ,or els the one is fet dirtRly again fie the other 3fo that if they be produced they fliouldmake one right line. As for example * H B Suppofe that the lines C D and £ F be parallels to one and the felfe fame line A B and let them not be parallels the one to the other. Then I fay,that the two lines CD & £ £,are diredly fet the one to the other. For for as much as they are not parallel lines, A letthemconcurreinthe point G, And from the point G draw a line cutting the line AB in the point H.Now by the former propofition the angles AHG &.HGC are equall to two right angles,but by the fame propofitio, the angle — A H g,is equall to the alternate angle H G f* Wherefore G the angles HG C and HG Fare equal to two right angles. Wherefore (by the iqpropofitionJthelinesC’G’ and FG are drawen diredly and make one right line. Wherfore al¬ fo the lines C D and £ F are fet diredly the one to the other: and being produced they will make one right line. G E r $&The lo.Trobleme, The luBropoJition* By a point geuen Jo draw vnto a right line geuen ^ a parallel line . ^^^Tppofe that the point geuen be ytfind let the right line geuen be e fg q it is required by the point geuen yiamely Jffio draft Vnto the right line © C?a parallel line. Take in the line C a point at alladuentures 9 and let the fame be D.and (by the firft peticio)dralb a right line from the point A, to the point D.And (by the 2 3 .propofition) Vpon the right line geuen A T)}and to the point in it geuen A^make an angle VAE^quall to ^ v 0 the angle geuen A VC. And (by the 14 , propofition) put vnto the line yf E the line A FdtreHly^ in fuch forte that they LJUi both " Parallellines are ^nderfian* ded to be fro dan¬ ced infinitely. Gwflru&it*. Deimnftrathn. TheJirB^Boo^e both make one right line. And fora/much as the right line A D falling Vpon the right lines ft C and E F,doth make the alternate angles ,namely,E AD tand A IDC equally one to the other, ther for e(by theiy,propofition)EFis a parallel line toft C.Wherfore by the point geuen ynamely A,is draWne to the right line geuen ftCa parallel line E A F: which "teas required to he done „ This proportion is to be vnderftandcd ofa point geuen without the line ge- uen,and in fuch forte alfo, that the fame line geuen being produced, doo not fail vppon the pointc geuen* The n.Theoreme. The izfPropofition* ffoneofthefydes of any triangle be produced: the outwarde angle that it ma\eth,is equal to the mo inward and oppofite angles. ^And the three inwar de angles ofa triangle are equall totWorwbt anvles. Vppofej A ft C he a triangle, produce one of yfides therm of namely yC ft to the point e ID. Then Ifay , that the outWarde angle A CD is equallto the two inward e and oppoflte angles C Aft & AftCiandy the three inwar de angles of the triangle , that is, the angles A ft C,ftC A,and C A ft are equall to two right angles. For(by the propofit ion going before )rayfe Vpfro the point C,a parallel to the right line A b ft, and let the fame be C E.Andforafmuch as Aft is a parallel to C E,and Vpon them fade th the right line ACx therefore the alternate angles ft AC and ACE are equaU the one to the other , Agayne forafmuch as A ft is a parallel Vnto C E,and Vpon them faUeth the right line ft D,tber fore the outward angle EC D is (by the zy.propofition) equallto the inward and oppoflte angle A ft C. And it is proued that the angle A C E is equal to the angle ft AC: wherfore the whole outwarde angle A CD is equall to the two inward and oppoflte angles, that is, to the angles ft AC and A ft CJPut the angle AC ft common to them both,VFherfore the angles AC D and A C fttare equall to thefe three angles AftQftC A,and ft AC. ftut the angles AC Do* AC ft are equall to two right angles (by the ij. propofition)iwherfort the angles A C ft, C ft A, and C A ft are equall to two right angles . If t her fore one of the fides of any triangle be produced,the outward angle that it mahthfls equad to the two inward and oppoflte angles. And the three inward angles ofa triangle are of Euclide $ Elements s . Fol.^u are e quail to ffro right angles: Tfrhicb was required to he demonfirated, Euclide demonftrateth either part ofthis compofcd Theoreme,by drawyng fro one angle of the triangle a parrallel line to oneofthcfides of the fame triangle, withoutthe triangle. Either part therof may alfo be proued without drawyng of a parallel linewithoutthe triangle, only chaunging the order of the thinges re¬ quired or conclufious.For Euclide firftproueth that the outwarde angle of atri • angle(oneofhisfides beyngproduced)is equallto thetwo inwardeand oppofne angles; and by that he proueth the fecond part: namely,that the 3 .inward angles o fa triangle are equall to two right anglesJBut here it is contrariwile. Forfirft is proued that the three inward angles ofa triangle are equallto two right angles, and by that is proued the other part oftheTheoreme, namely, that one fide ofa triagle beyng produced, theoutward angle is equal to the two inward and oppo¬ fite angles,And that after this maner* Suppofe that there be a triangle ABC, and produce the fide BC to the point E. And take in the line B C at al a point auentures which let be F : &drawalinefroin^toF.AndbythepointFdrawe A n vnto the line^S a parallel line (by the former pro- pofition)which let be F D, Now forafmuch as F D is a parallell vnto A B, and vpon them falleth the right line AF,and alfo the right line2?C,therfore the alter¬ nate angles are equalhand alfo the outward angle is equall to the inward angle.Wherefore the whole an¬ gle A F C is equall to the angles F A B and ABF. And by the fame reafon(if by the point F we draw aparal- lel line to the line A C) may we proue that the angle A F B is equall to the angles F A C,and A C F . Wherfore the two angles A FB 8c AFC are equall to the three angles of the triangle A B C.But the two ang lesAFB 8c AFC are(by the 1 3 .prop ofition)equall to two right angles, Wherfore alfo the three angles of the triangle ABC are equall to two right angles. But the angles ACF and <^ACE are alfo (by the 13. propofition) equall to two right angl es .Take away the angle A C F which is common , w herfore the an gle remai- ning,namely,the outward angle ACEis equall to the two angles remaining, namely, to the two inwarde and oppofite angles A B C and C A B ; which was required to be proued. Eudemus affirmeth that the latter part ofthis Theoremc , The three angles ofa triangle are equall tot\\>oright angles, was fir ft foundout by Jp thagoras^whofe demon; ftration thereof is thus. Suppofe that there be a triangle AB Crand by the points, draw (by the former propofition) vnto the line £C,a parallel line, which let beD£.And forafmuch as the right lines -BCand-DfEare parallels, and vpon them falleth the right lines A B and./4C,therefore(by the 2p. propofiti- T> onj the alternate angles are equall. Wherefore the angle D A B is equall to the angle ABCran<X the angle £^Ctothe angled CB. Adde the angle B A C common. Wherefore the angles DAB, BA C,CA E , that is, the angles DAB and# B v4£,namely,two angles equal to two right angles^are equal to the thre angles of the triangle^ B C. Wherfore the thre angles of a triangle are e- quail to two right angles: which was required to be proued. The conuerfe ofthis propofition is thus* Mq. If sth other de- manffratsea* The latter part ofthis Theo • remejirji found tut bj IUthaga-. ras. T he demanftra* tion thereof af¬ ter him » The canuerfeof shit props fit ton. Demonfiratiom cf the firfl part •fthe canuerfe . Demonfiration : cf the fecond part of the con- oterfe. 4 Cerrollarj. Euerj right li¬ fted figure is re¬ fitted in tri¬ angles, A triangle is the fir (l of all fi- gures. Into hove many triangles a fi¬ gure may he rc- feltttd. 7 he firfl Boo{e If the outward angle of a triangle be equall to the tWo inward angles oppofite again ft it : one of the fides of the triangle is produced, and the line without the triangle, is dravten direttlj to the fide of the triangle, & maketh one right line With it. And if the thre inWard angles of a rettiline figure be equal to two right angles,thefkme rettilinc figure is a triangle » Suppofe that there be a triangle ABC: and let the outward angle A CD be equal to the two inward & oppofite angles A B Cand CAB. Then I fay that the fide BCis produced to the poynt D, And that -5C2) is one right line,For forafmuch as the angle ACD is equal to the two inward & oppofite angles, addc the angle ACB common. Wherefore the angles ACD and ACB are equal to the three angles of the triangle A B C .But the three angles of the triangle ABC are equall to two right angles Wherefore alfo the two angles ACD and ACB are equall to two right angles . But if vnto a _ right line,and to a point in the fame line be drawen two B right lines, not both on one and the fame fide, making the fide angles equal to two right angles :thofe two rightlines fhal be drawe dire&ly, and make one right linefby the H-propofirion.^Wherefore the right line BC is dra¬ wen dire&ly to cue line C D, and fo is B CD one right line; which was required to be proued. Agayne fuppofe that there be a certayne reftilincfigure AB C,hauing onely three ang'es, namely, at the pointejM^C: which angles let be equal to two right angles Then I fay that *ABC is a triangle-Firll^C wrohe right line . For draw the line2? D. And forafmuch 4s in either A of the triau gles A B D and D B C, the three angles are e- qua! to two right angle s,of which the angles at the points e//,B,C,are equal to two right angles. Wherefore the an¬ gles remayning, namely, ADB and CDS are equall to two right angles. Whereforef by the 1 4-propofition ) the line D Cis let dire&ly to the lineD A, Wherefore the fide AC is one right line.And in like fort may we proue that the fide zABis one right line,and alfo that the fide BCis one right line. Wherefore the figure eA B Cis a triangle: which was required to be proued. By the fec5d part of this 29,propofitio, namely fhree angles of a triangle are equall to tworight angles, may ealely be knowen , to how many right angles, the angles within any figure hauing right lines and many angles are equall. As arc figures offower angles, of fiue angles,offixe angles, and fo confequently: and infinitly* And this is to be noted , that euery rightlined figure is icfolucd into triangle. For that a triangle is the firfl: ofall figures . For two lines accomplifh no figures V Vherfore how many fides the figure hath, into fo many triangles may it be rc- folued,fauing two.As if the figure hauefowerfides,itis refolucd into two trian¬ gles, if ithauefiuefides,into5.triangles:if6 fides into 4, triangles, andfo con¬ fequently ,and infinitly .And it is proued that the three angles ofeuery triangle are equall to two right angles. VYhereforeifyou multiply the number of the triangles, into which the figure is refolued, by two>youfhall haue the num¬ ber of tightangles , to which theangles of thefigure are equall. So the angles of euery qua irangled figure are equall to 4,right angles. For it is compofedoftwo triangles. And the angles ofafiue angled figure are equal to 6-rightangles, for it is compofed of three triangles ,andfo forth in like order. The redieft andapteftmanerto reduce any rc&ilinc figure into triangles, is thus ofEuclides Elementes , Eol\i, thus.From any one angle of the figure to euery other angle (of the fame) bey iig oppofite vnto it, drawe a right liue,fo (hall you haue all the triangles of that fi * gure described* In a quadragle, from one angle you can drawe butonelyne to thcoppofice an gle,by which it is deuided into two triangles only. In a pen¬ tagon figu re, from one angle you may draw lines to two op polite angles, and fo you fhai haue three triangles, In an Hexhgon, you may from One angle draw lines to thre oppofiteangles,and fo fliall you haue 4. triangles. In an heptagon, from one an¬ gle may be drawne lines to foure oppofite angles, and fo fhal there be flue trian^ gle. And fo confequently ofthc reft* As you fee in the figures here fet. This thing may alfo be thus cxprefled.In any figure of many fides, the num- Zotnbwrhilii- ber of the angles of the figure doubled, is thenuber of the right angles to which ZerofrightZZa* the angles ofthe figure are cquall/auing* foure. As for example* gfa 9itfo which Let there be an hexagon figure ABODE F,and within it take a point at all the«”Sl“ °f '*• auentures,namely,G* And draw from the fame point figure*™ w to euery one ofthe angles a right line,8c fo dial there be comprehended in the figure fo many triangles, as there are angles in the fame. Whereforeby this 32,. propoficion all the angles of thefe triangles taken toa c gether,arc equall to double fo many right angles, as there beangles in rhefigure* Wdierefoteforafmuch as there are fixe triangles, there are tweluc right an* glcs .Butall the angles at the point G arc equall to 4, right angles by the i^propofmon. Wherefore rake away foure out oftwelue,and there reft eight, Wher fore thefixe angles in the Hexagon figure are equall to eight right angles. By that which hath now bene declared, it foloweth that all the angles of any fi- gure hauing many fides, take together, are equal to twife fo many right angles, mcZcJJZj, as the figure is in the reaw or order of figures.A triagle is the firft figure in order* &,his angles are equal to two right angles .which are twife one. A quadrangle is AtrUnsle *he the fecond figure in order* Wherfore his angles are equal to fower right angles ** which are twife two.The order offigures is gathered ofthe Tides, For if you rake ^ ^draHle two from the number ofthe fides of a figure, the number of thefidcs remayning, the fcco”d> an* is thenumber oftheorder ofthefigure. As if you will know,how many in order is a figure of fixe fides: from fix(which is thenumber of his fidcs)take away two* offigures hr* and there will remains foure. VVherforca figure offixe fides is the fourth figure thsredv. M.u* in A B other Cor - VelUtj. i n the order of figures .Then double fourc/o {hall you hauc cigh r. Wherefore the angles therof arc equal! to eight right angles. And fo of alLoth cr figures. Hereby alfoitis manifeft, that the outward angles of any figure of many fides taken together, are equall to fourc right anglcs.For the inwardc angles together with the outward angles ,are equall to twife fo many right angles, as there be an¬ gles in the figurc(by the i3.propofition)Bnt the inward angles are equal to twife fomany rightangles, asthere be angles in the figure, fauyng foure: as it was be¬ fore declared. VVherfore the outwardangles are always equal to foure right: ana gles* As for example. Suppofethat there be a pentagon3A BCD E.And produce the fiue fidcs ther- -ofto the points FjG^jKjL.Nowfby thei^. propofitio)the two angles at the point A fiiall be equal to two rightangles .And(by the fame) the two angles at the pointe B fhall be alfo e- quall to two right angles* And fo taking eucry two angles, they fliaii be in all equall to tenne right angles . .V Vherforc taky ng away the in¬ ward angles , whiche(as hath before bene pro- ued) are equall to fixe righteangles, the out¬ wardangles (hall be equallto fower right an* gles.Andfo of all other figures. other Ctr- ^ 15 a^° manifeft»that euery pcntag5,which is fo deferibed .thatech fide therof * deuideth two of the ocherfidesjhathhisfiueangies equall to two rightangles. For fuppofe that ABC DE : be fuch a pentagon as is there required fo that let the fide AC cut the fide B E in the point G-.&c let the fide AT)^ cut the fame fide B £ in the point F. Now the by this propo- fitiontheangle^F(7fhalbeequalltothe two angles at thepointFand^D:namdy,theoutwardangletothetwo ® inward and oppofite angles. And by the famereafon the angle F G A is equal to the angles at the points C and E which are in the triangle C£<?.But the two angles AFG and F G ^together with the angle at the point A, are e- quall to two right anglesf by this propofition ). Where¬ fore the fower angles at thepointes, B,C, £>,£, together with the angle at the point A, are equal to two right an¬ gles .-which was required to beproued. jtn other Cor- tralLrj. <An ether Cor- O'xMerjs By this propofition alfo it is manifeftjthat cuery angle ofan equilate trian'- gle is two third partes ofa right angle.And that in a triangle of two equall fides hauing aright angle at the toppe, either ofthe two angles at the bafe is the halfe of a right angle. And in a triangle called Scalenum.fuch a Scalenu ( I fay ) which is made by the drawght ofa perpendicular line from any one oi the angles of an equilaier triagleto the oppofite fide therof, one angle is a right angle,an other is two third parts ofa right angle, namely , that angle which was alfo an angle of the equilater tnanglejwherforeofneceffity the angle remaining is one third part of a rightangle.For the three angles ofatriaglc mull be equall to two rightangles* Moreouer by this propofition it is manifeftjthat if there be two triangles, and if two angles of the one be equal to two angles of the othenthc angle remai- niog mng {lull alfo be equall to the angle remay mng.F or fowCrauch as three angles of any criaugle are equal to three angles ofany other triangle (for that in ech the three angles are equal to two right angles)- If from ech triangle be taken away the two equall angles }the angle remay mng fhall (by the common lentence)be equallto the angle remay mng. And here I thinke it good to (hew how to detiide a right angle into three c- quail parteSjfor that the demonftration thereof depended! of this proportion. A Suppofe that there be a right angle C^contayned of the right lilies AB and B C\& in the line B C,take a point at all aduentures, which let be D.An d vpon the line B ‘D deferibef by the firft)an equila- ter triangle B E> £,And(by the p.propofition)deuide the angle D B E into two equall partes by the right line B F. Then / fay that the right angle A B Cis deuided into thre eq ual parts by the right lines B E and B F.For forafmuch as £ F D is an equilater triangle,therfore as hath before bene declared, the angle SB D is two thirdc partes of a right angle. B ut the whole angle ABC3 is a right angle. Wherfote the angle remaining, namely,^££ is one third part of a right angle. Again forafmuch as the angle EBD, is two third partes of a right angle,andit is deuided into two equall parts by the ri^ lne B £3therefore either of thefe two angles E B E 8cF BT)is; one third part of a right angle.Wherefore the three angles -<42? £,£2? £ and 2) are equall the one to the o- ther. Wherefore the right angle A B Cis deuided into three equall partes by the right lines B E and B F: which was required to be done. 1'he %\fTheoreme. The TftfPropofition* Two right lines ioyning together on one and the fame fide, mo equall parallel lines: are alfo themfelues equall the one to the other >and alfo parallels • fppofe that A 3 and CD he right lines equal \ and parallels : and let thefe Vtoo right lines Cand 3D ioyne the together > the one on the one fide 3and the other on $ other fide -Then I fay that the lines AC i?3 Dare both equally alfo parallels. S)raH? (by the firfl petition) a right line from the point 3 to thepoint Ct And fora afmuchasA3 is a parallel toCD, and \>po them falleth the right line 3 C, tber* fore the alternate angles A 3C and 3 C P are equall the one to the other (by the %9.propofition). And forafmuch as the line A3 is equallto the line C D3and the line 3 € is common to them both , MJiL there* ffov> t» fUttsde# right angle int $ three equall ThefirtlTfiookp therefore thefe Wbo lines AB and B Cyare equall to theft tSbo lines ) BCandC 3j,and the angle A B Cis e quail to the angle B C DyVberfore (by the 4.pro* propofiuimfihe bafe B D is equall to the bafe A C,and the triangle A B Cfi e- quail to the triangle B C T>yand the angles remayning are equall to the angles remay ningthe one to the other J>nder Tbhich are fub tended equall fide suffers* fore the angle AC Bis equall to the angle CBD , and the angle B AC to the angle B T) CAnd forafmuch as y>pon thefe right lines A C and B Dfalletb the right line B C making the alternate angles ,that is the angles ACB and CBD , equall the one to the other ^therefore (by the zy.propofition) the line ACisa par ailed to the line B tD.Jfnd it is prouedthat it is equall Vnto it . Wherefore two right lines soyning together on one and the fame fide two equall lines which are parallels ,are alfo themjelues equall the one to the other, and aljo parallels: Tbhicb 'toas required to be proued. The i^.Theoreme. T he 34. Tropo/ttion. fn farattelogrammesjhefides and angles which are oppofae the one to the other , are equall the one to the other* and their diameter deuideth them into two equall partes. BVppofe that A BOD be a parallelogramme and let the diameter ther * of be B C.Tben I fay that the oppofitefides and angles of the paralhlo » gramme ACDB are equall the one to the other yandj the diameter BC deuideth it into Wbo equall partes . For forafi much as A B is a parallel line bnto C fyand V* pon them falleth a right line B C: therfore (by the 2 <y,propofition)the alternate angles ABC and BCD are equall the one to the other . A • gay ne forafmuch as AC is a parallelline to B lD,and Vppon them falleth the right lyne B C: therfore (by the fame) the alternate angles, that is, the angles ACB and CBD are equall the one to the other . Nolb therfore there are tV>o triangles ABC and BCDfhauing Wbo an - gles of the oney namely % the angles ABC and ACB equall to Woo angles of the other that isyto the angles BCD andC B D, the one to the other jmd one fide of the one equal to one fide of the otherjiamelyjhatfydethatlietb beWbene the equall angles Jbhich fyde is common to them both, namely, the fide B C, Wber* fore (by the 26. proportion) the other fides remaining are equall to the other fidesremamingghe oneto the other ,and the angle remaining^ equal to the an* gle rmaynmgyVhexfore the fide A B is equall to the fide CD,md the fide AC ofEuclides Ekmentes . to the fide B Dgs* the angle B AC is equal to the angle B D C.And forafmucb as the angle ABCis equal to the angle B C Dtand the angle CBDto the an * gle AC B : tberfore (by the fecond common f entente) the whole angle A B D is equall to the whole angle A C Dt And it is proued that the angle B AC is e* quail to the angle C D BjAClurfore in parallelogrammes } the fides and angles which are oppojite are equall the one to the other, I fay al(o that the diameter therof deuideth it into two equall partes. For forafmuchas A B is equall toCD} and BCis common to them bothgherfore thefe two A Band BCare equall to thefe two BCandCDtandthe angle A B Cts equal to the angle BCDyFher fore (by the ^.propofition) the bafeACis equall to the ba fe B D>and the t*i* angle A B Cis equall to the triangle B C ID. VFherfore the diameter B C den uide tb the psralldogramme A B CD into two equall partes : which is all that Was required to be proued. In this Theorerae,are demon Crated three paffions or properties ofparallei? logrammes. Namely, that their oppofite files are equall; that their oppofice an? gles are equall-.and that the diameter deuideth the parallelogrammc into two e- quall partes* Which is true in all kindes ofparallelogrammes. There arc fo wer kindes of parallelogrammes ,a iquare, a figure of one fide longer then the other, a Rhombus,or diamond figure,and a Rhomboides or diamondhke figure. And here is to be noted, that in thofe parallelogrammes, all whofe angles ar right an- gles(as is a fquare,and a figure on theone fide longer) the diameters do notonly deuide the figure into two equall partes,but alfo they are equal the one to the o* ther*As for example. Suppofethat^SCD beafquare, 4 B A B or a figure on the one fide longer, and draw in it thefe diametres A D and 5 C.And forafmuch as the line >4 5 is e- quallto the line CD (by the definitio of a fquare,and of a figure on the alfo one fide loger)& the line A Cis com¬ mon to the both ; therfore two fides of the triangle A 5 Care equal to two fides of the triangle A CD gat one to the other, and the angles which they a contayne are equall, namely, the an- C v gles 5 AC & AC Z>, for they ate right angles. Wherefore the bafes namely, the diame¬ ters A D and 5C,arefby theq.propofitionjequal, Butin thofe parallelogrames whofe angles arcnotrightangles,as is a Rhom¬ bus ,and a Rhomboides, the diameters be euer vnequall. As for example. Suppofe that ABC D be a Rhombus,or a Rhombaides and drawe in it thefe diame¬ ters A C and 5 D. And foras¬ much as >45 is equall to C2>, and B Cis common to them both,&the angle>45 Cis not equall to the angle 5 C D ( by the definition of a Rhombus and alfo of a Rhombaides) M.iiii. B T tire fastens of far alio tegrames demo ft rated in this T horeme. Power kjndes of faraHelo- grammet. ’ ■ ‘-f-. there *The eonuerfe of this prep opt ion itfter Vrqclm, A Corollary ta¬ ken emt of v’ ThefirslTiooke therefore (by the 24,propofition) the bafes alfo are vnequall, namely, the diameters zACa.n6.BD. Agaync.Inparallelogrammesofequallfides,asareafquare, anda Rhom* bus, the diameters do notonely delude the figures into two equall partes, but alfo they deuide the angles into two equall partes* For fuppofe that there be a fquare or Rhombus AB CD, and draw the diameter e/tf D. And forafmuch as the fides A B and B D are e- quall to the fides A C and CD(for the figures are equilateral and the angles sAB D and^ CD are equally for they are oppofite angles ) and the bafc ADisrommontoboth triangles. Therefore (by the fourth propofition)the angles BAD & CAD are equail,and fo alfo are the angles BDA and C ‘D A equall. Wherfore the angles i? AC and CD B are deuidedinto two equall partes. Butin parallogratnmes whofe fides arenot equall, fuch as area figure on the one fide longer, and a Rhomboides it is not fo. For fuppofe AB CD to be a figure on the one fide longer or a Romboides, And draw thedia- A B IB meter zA D,And now if the angles B A Cand CD *S,be deuidedinto two equall partes by the dia- meter-^D,theu forafmuch as the angle CAD is (by the 2<?.propofition,) equall to the angled *D .8, the angle alfo BAD fhal be equal to the an¬ gle A D B(by the firft common fentence), Wher¬ fore alfo the fide is equall to the fide B Dfby the tf.propofitio ). But the fayd fides are vnequal: which is impoifible.Wherefore the angles B AC c and C D B are not deuided in to two equall partes* The eonuerfe of the firft and fecond part of this propofition after Proclus. if a reel ihne figure whatfoeuer haue his oppofite fides andangles equall : then is aparallelograme. For fuppofe that AB CD be fuch a figure,namely,which hath his oppofite fides and angles equall. And let the diameter thereof be zAD. Nowforafmuchas the fides A rB and 2D are equall to to the fides'D C and zA C, and the angles which they co~ A_ tayne are equall, and the bafe AD is common to ech tri- angle,thereforef by the 4,propofition Jthe angles rema y- ning are equall to the angles femayriing,vnder which are fubtended equal fides.Wherfore the angled AD is equal to the angle A D C, and the angle zA D B to the angle CA D. Wherefore (by the 27. propofition ) the line ABisa parallel to the line CD^and the line AC to the line B D, Wherefore the figure AB CD is a parallogramme:which was required to be proued. \ / \ ! ^ A Corrollary taken out of Fluffates. tA right line cutting a parallelogramme Which Way foeuer into two equall partes , Jhall alfo de¬ uide the diameter thereof into two equall partes. For of Euclides Elementes. FoL 4.?. For if it be polfible let the right line G C deuide the parallelogramme A SR JO into two equal partes, but let it deuide the diameter D E into two vnequal! partes in the point h And let the part / £ be greater then the part / Z>*And vnto the line ID put the line/O equall(by thg. propofitioj. And by the point 0,draw vnto the lines -4 2> and-# S apa- rallelline O £(by the 3 1 .propofition .) Where¬ fore in the triangles EG I and CD I, two angles ; of the one are equal to two angles of the other, namely, the angles IOF and I DC ( by the 2p* propofition),& the angles FI O 8cCID( by the 1 5 .propofitio),& the fide ID is equal to the fide I O. Wherefore (by the 2 ^.propofition ^the tri¬ angles are equall.Wherefore the whole triangle Sigis greater then the triangle D IC. And forafmuch as the trapefium GBDCis fup- pofed to be the haife of the parralleIograme,and the halfe of the fame parallelograme is the triangle EB D (by this propofition) .From the trapefium GB DC and the trian- fle££-Dwnichare equalhtake away the trapefium (?£ D/whichis common to them oth,and therefidue namely, the triangle D IC lhalbeequall to the refidue, namely, to the triagle £/(7:butitis alfo leflefas hath before ben proued): which is impoffible. Wherefore a right line deuiding a parallelogramme into two equal! partes, {ball not deuide the diameter thereof vnequally. Wherefore it fhall deuide it equally ? which was required to be proued. An addition of P elitarius* Betty enettyo right lines being infinite and making an angle geuen : to place a line equall to a line geuen, in jack forte, that it Jhau make Veith one oft ho/e lines an angle squall to an other angle ge* utn,N°ty *t behoueth that the ttyo angles geuen be le/fc then ttyo right angles. Suppofe that there be two lines A B and A C,making an angle geuen BA C: and let them be infinite on that fide where they open one from the other* And let the line geuen beD, and let the other angle geuen be £. And let the two angles A and £ be lefle then two right angles ( otherwife there coulde not be made a triangle,asitis manifeft by the 17 propofition) . It is required betwene the lines csf B and AC to place a line equall to the line geuen D, which with one of them as for exaple with the line A C, may make an angle equal to the angle ge ^ uen £.Now then vpon the line C and to the pointinitcxf.make an angle equail to the angle geue£(by the 2 3 .propofition), which let be CA £,And produce the line FA on the other fide of the point A to the point G:znd let A (/be equall to the line geuen D ( by the 3 . propofition ). And by the point (?,draw(by the 3 1 .propofition) a parallel line to the line oAC, which let be G H, and produce it vntil it concurre with the line A B: which concurfe let be in the point//. And agayne by the point// draw the line HK parallel vnto the line (//.-which let cut the line exf C in the point ZT.Then I fay that the line H K is placed betwene the lines AB 8c AC & is equall to the line D*And that the angle at the point K is equall to the angle geuen £.Forforafmuchas(byconftru&ion)-4<?///:is a parallelogramme the line K His equall to the line A (/(by this propofition ). Wherefore alfo it is equall to the line D. And forafmuch as the line A ZTfalleth vppon the two parallel lines, F G and K //, therforc the angle A K His equal to the angle F A K( by the 2p.propofitio.) for that they are alternate angles-Wherfore alfo the fame angle at the point £is equal to the angle geuen S. Wherefore the line Z/K being placed betwene the two lines AB and <sA C,and being equall to the line D,maketh the angle at the point K equall to the angle geuen £: which was required to be done. Though this addition of Pclitarius be not fo muche pertayning to the N.i* propo- Hemon/lratim leading to tt ft ah [nr dy tie. An addition of Vclitarittt. CenJlruHsm. Demmftrarim Demtnftrdthn Three cafes in this fropofttion„ Thcfirjt cafe. propofitiomy ctbecaufeit is witty and femcth fomewfm difficult, I thought it good here to ancxe it. 4 if ' —*'• l ■ ■ . * . f 44J,. v r . Theiy .Theorems. The tf.Tropofoion. TaraUelogrammes confi/lingvppon one and the fame bafe, and in the felfe fame parallel lines, are e quail the one to the other. - p Vppofe that the/e paraUelo * f if grammes A 'B CD and EB 4 CF Jo confeft vpon one and the fame bafe /hat iss Vppon & J J 3 1 If B Ctand in the felfe fame pa rallel lines y that is J F,and B C.Tbeti 1 fay /hat the parallelograrne ABC Bis equal to the parallelograrne EEC F.For forafnuch as A $ CB is a pardUetagramme, ther* fore (by the $ 4 .propofttion ) the fde A &7is equallto the fideB C, and by the fame reafm alfo the fide EE is equall to the fide BC/vhcrfore JED is equall to EFandB Eis common to them both- Wherfore the tbbole line A E is equall to the whole line/D FsAnd the fide Al 8 is equall toy fide D Cypher fore the/e two EA and AB are equall to theft Vtoo HD and DC /he one to the otheriandy angle FDC is equall to the angle EABy namely/he outward angle toy inlrard angle (by y n)tpropofitio):ts>herfore(by jt 4 propofition)the bafe E B is equall to the bafe ECy and the triangle EA Bis equallto the triangle FED C.Take ale ay the triangle EDGE, lehich is common to them botb.VVherefore the re ft due, namely, the trapefium A BG Bis equall to the re/idue, that is Jo the trapefium EG C FfPut the triangle GB C commo to them both Wherefore the to hole paralleUgramme ABC Bis equall to the "tobole parallelogramme EB C FVVherefore paralletogrammes con f fling vp* on one and the fame bafe, and in the felfe fame parallel lines yare equal the one to the otber.'fchich was required to be demonflrated. Parallelogrammcs are fayde to be in the felfe fame parallel lines, when their bafes,andthe oppofite Hies vnto them, are one and the felfe fame lynes wy th the parallels* In this propofition are three cafes* For the line B E may cutte the line AF, either beyond the point D,or in the point D,or on this fide the point D * When ofEuclides Elementeu FoL\6. itcnttcth the line A F beyond the point D the demontetion before put fcr^ ueth* V: .• e • ■ ' ••• - - : • . * \ * * V i \ ' * > Butiftheline££do cutte the line AF in the point D,then forafmuch as ( by the former propofition)the triangle £ CED or 2? C£ is the halfe of either of thcfe paralelogrammes ABC ‘Da.ndEBCF ( for intheparallelogramme^ B CT> the diameter B D maketh the triangle B DCthe halfe of the fame paraIlelogramme,and in the parallelogramme EBCF the diameter £ Cor DC maketh the felfe fame triangle £‘D Cthe halfe of the parallelogramc££ CF Jther* fore(by the 7. common fentence)thcparallelo- grammes A B CD and £ B CF are equall. But if the Hue BE do cutte the lyne *AF on thi5 fi.de the point D, then forafmuch as ei¬ ther oF the lines AD and EFis equall to the line B C,therefore by the fir ft common Sentence they are equall the one to the other. Wherefore taking awav ED, which is common to both y the refid ueV^ £ fhalbe equall to the refidue.D F. Agaync forafmuch as(by the ^.propofitio) the line zA B is equall to the line C D, and (by the 17. propofition)theangle£ A B is.equaUo the angle FDC: therfore (by the 4. proposi¬ tion) the triangles E 2? A and F C D are equal* Adde the trapefium CDEB common to them both : and fo (by the feConde common Sen¬ tence jthe two paraflelogrammes */{ B CD and ££C£fhaIbe equall : which was required to beproued. The/ecmdeitfi* The third ca ft. h&'The i6:Theoreme. 'ThetfSPropofition. ParaUelogrammes confi fling vpon equall bafes, and in the felfe fame parallel lines ^are equall the one to the other. Vppofe that thefe parallelogramtnes AB CD and EE G H do con ft jl ypon equall ba/es3that is^pon BCandF G,and in the felfe fame paraU ■ lei lines 9t bat is 3A Hand I B G, Then I Jay %t bat the parallelogramme A BCD is equall to the parallelogramme EFG FLDra'Se a right line from the C9”J?ruaio*‘ point B to the point Eyand an other from the point C to the point H. Jndfor* Demorfr#** afmuch asBC is equall to F GJhut F G is equall to E H, therfore <B Calfo is e* quail to E H3 and they are parallel lines , and the lines B E andC B ioyne them together : hut npo right lynes ioynyng together tlt>o equall right Hdj* lines “V ■l Three cafes in this propofttion. The fir ft cafe. Eaerj cafe maj happen fetiea timers wajet. a. 3» % •*».*. <f7 •Ji-aff.’.r' E H 7 : C A*1 y ’ / ■ : - Jr / sj / '1 -- *. if •- ■ 1 if 7 if ... . ' . '! / j / ■ ;i Jr / j fine s being parallels; art themfeluts alfo (by the ^ proportion) e quail the one to the other , and parallels, VVherforeEB CH is a parallelo* gramme 3and is equall to the parallel lograme ABCS): for they haueboth one and) fame bafeythat tsfB C,And are iny felfe fame parallel lines ythat is3BC& EHtAnd byy fame reafoit alfo the parallelograme EFG H is e* qual to the parallelograme E B CH ; VVber fore the parallelograme AB CS) is equal to the parallelograme E & c: ^ FGH, VVher fore parallelogr antes conjifting vppon equall bafes3andin the felfe fame parralkl lines, art t quail the one to the other.Tahich 'toas required to beproued. i,. .?• -. . , j . ^ ^ , _ f. ( , . ( „ In this propoficton alfo are three cafes.For the equall bafes may either be v t¬ terly feperated a fonder: or they maytouefoeat oncof the codes: ortheymay haue one part common to them both. Ettclides demonftration ferueth when the bafesbe vtterly feperated a fonder* Which yet may happen feuen diners waycs.For thebafes being feperated af5*> der, their oppofite fides alfo may be vtterly feperated a fonder beyond thepoinfi D,as the Tides A D and E H in the firift figure. Or they may touche together in one of the endes, and the vhole fi demay be beyond the point D,as the Tides A DandE H do in the fecond figure. Or one part may be beyond the point D, andan other part common to them 4 b cp d B C F <3 3 C F both ,a s in the third figure,the fides A D and E H haue the part E D common to them both* Or they may iuftly agree the one with th'cother,that is,the pointes A and D may fall vpon the pointes Eand H: as in the fourth figure. Or the fide A D being produced on this fide the point A}part ofthe oppofite y. fide vnto the bafeF G m'ay be on this fide the point A, and an other part may be common with the line A D,as in the fifth figure* Or one ende ofthe fide EH may light vpon the pointcA3 and the whole fide 6. on this fide of it: Asinthefixt figure. Or the faid fide E H may vtterly be feperated a fonder on this fide the pointc A, as in the feuenth figure. And the two other cafes affo may inlike mancrfiauc feuen varieties: as in th„e figures here vnderncth and on the other fide of this leafefet it is inanifeft. and here is to be noted^thac in thelft three cafes and in all their varieties alfo, the conftru&ion SC demonftration put by Eqclidef namely ^thc drawing of linos fro the point B to the point E Sc from the pointe C to the point H,and fo prouing ic by theforinetpropofition) will feme oncly in the fourth varietie ofech cafe, there nedeth no farther conftructiomfor that the conclufion ftraight way folio? weth by the former propofition. The like *varie* tj in ech ofthe other two cafes , Each da con - Jirullian and demonftration feructhinall thefe cafes, and in their9aritiet alfo. A "DE H & F C <3 A B D H AR T>ff 3 F C <3 » F O G Canftntftion, Demonftratiott. i 1 '■ Cj . ; l\ TheijaTheoreme . 7 be ty.Tropo/ition. ■ ", ■’ r ‘ i \ , ** 41 . i T riangles con f fling upon one and the felfe fame bafe?and in the felfe fame par allesiare e quad the one to the other* Vppofe that the re triangles A BCandlDBC doconfift Vpon one and the fame bafe>namdyi B Cyand in the felfe fame parallel lines jhat is , jiT> and B C. Then I fayy that the triangle ABCis e quail to the triangle B(DC. Produce (by the 2. peticion ) the line At) on ech fide to the pointer E and F. And (by the 31, proportion) by the point Hydras Ynto the line C A a parallel line B E and (by) fame) by the point C, draw Ynto the line B !Da pa* rallel line C F Wherefore EBCA. and D B C^are paraHelogrammes „ And the paralklogramme E B fAy is (by the $ 5. propofition) equal/ to the parallelogramme (DBCF , For they confift Yppon one and the (elfe fame bafe , namely , BCy and are in the felfe fame parallel lines , that isy B ( and E F, But the triangle ABf id (by the $4. propofim)tbt halfe of the parallelogramme EBCAyfor the diameter AB deuideth it into two equall par ts:&( by the fame)thetri • £ C angle BBC is the halfe of the pa a rallelogra mmelDBC F for the diameter B> Q deuideth it into two equall parts: but the balues ofthinges equallarealfo equall the one to theotber (by they . common fentence ) therefore ehe triangle ABC is equall to the triangle (DB (V therefore triangles confining Ypon one and the felfe fame bafe3and in the felfe fame parallelsiare equal l the one to the other : Tphicb Y>as required to be demonilrated. Thofe ofEuclides Elementes . FoL^.8* Thofc triangles are faide to be comayned rithin the felfe fame parallel lines, HowtrUngtes which hauing their, bafes in oneofths parallel lineSjhane their toppes in the %e/hffiife°fJLe Other. ' • 1 parallel lines, Hereas I prom i fed will I flicw out of PCocIas the com pari fori of two triad- comparifonof gles, which hailing their hies equal! 'jti&ttc the Safes and angles at the toppe vucj two triangles quail* Arid Aril: I fay that the vneqnall angles- at the toppe being, equall to two wh°^^tie! btl‘ejr right angles, the triangles flialbe cqualLAs for example, , - ^ffJand angles at the toppe art Suppofe that thefe two triangles ABC and DEF haue twofides ofthe onemarne- ly, A B ahd A C^equall to two fides of the Other, namely, to D £ and D F, eche to his correfporideni: fide, that is, ABtoD £,and ACtoD F, and let the bafe B C be greater v/hen the two then the bafe £ F : andfet the angle at the point tA be greater then the angle at the angles at the pointD.Butletthefaydeanglesatthe pointes ^4andD, \ toppes are ec^uaU be equall to two right angles. Then I fay that the triangles ABC andD ££ are equall. For fora/rauch as the angle's AC is greater then the angle £ D £,vpon the line £ D,and tothepointD defcribe (by the 23. propoikion) an angle equall to the angle £^TC;which let be E D G : and put the line D G equall to the line A Ci and draw a line from E to G,and an other from F to G; and produce the lines ED& F D beyond the poynt D to the pointes H and AT.Now for afmuch as the angle 'B AC is equall to the angle ED G* and the angles b A Cani E D F are equall to two right am gles,thereforethe angles EDG andEDF are equall to two right angles .But the angles EDG and K D G,arealfo equal to two rightangles.-take away che angle FDG com¬ mon to them both: wherefore the angle remayning EDF is equall to the angle femayning.G D K. But the angle ED' F is equall to the angle'H D K ( by the 1 5 .propofition)for they are hed angles. Wherefore theangle G D K isequall to the angle HDK .And forafmuch as in the triangle G D F the outward angle G D H is(by the 3 2. propofition) equal to the two inward and oppofite angles at the points G and £: which two angles alfoare(by the 5. propoiition) equall the one to the other: for the line DG is by confku&ion equall to the line c AC, namely, to the line DF. Wherefore the angle G D.H is double both to the angle at the point <?,and to the an¬ gle at the point P.Biit the angle G D His alfo double to the angle GDK (forthe an¬ gle G D K is proued to be equal! to the angle K D H) wherefore the angle at D G F is equal! to Che angle G D K : and they zie alternate angles. Wherefore (by the 27. propoiition) thelineD E isaparallel to the line £ £7. Wherefore the triangles GDE and FD E arevppon one and the felfe fame bafe, namdy, Z>£, and in the felfe fame parallel lines D E and GF .Wherefore by this propofition they are equall. But the tri¬ angle GDE is by conftm&ion equall to the triangle ABC. Wherefore- alfo the tri¬ angle D E F is equall to the triangle zABC : which was required to be proued. But now let the ariglesA A C and £ CD £ be greater then two right angles : & let When thej ats the angle at the point A be greater then the angle at the point D, as it was before The ireater th'**e I fay that the triangle ABCis lelfe thea-the triangle £>£ £. Let the fame conlfruaion r,ght angles. beherothat was in the former. And forafmuch as the angles B ACznd S'DF that is, the angles EDCj and £CD£are greaterthen tworight angles, but the angles EDG and GD A" are equall to two right angles: take away the angle FD G which is common to them both . Wherefore the angle remayning,namely,£D£ is greater the the amrie remayningpiamelyjthen GD A: thatis,the 'angle JCD H (which by the 1 5. propofition is equall to the angle £ D F)is greater then the angle G D K, wherefore the angle G D H is more then double to the angle. G.D K : but the angle f? D His double to the an- N.iii;‘. glc Whettthcj *rt leJJ'e then tn>a right angle t. cHbefirH(Boo^e glc DGF , as was before proued, Wherefore the angle GDK is leffe then the angle DGF. Vnto the angle GDK put(by the a 3 .propofition ) the an¬ gle OG £ equall: and produce the line G L till it concurre with theline ££inthepointe L. And draw a line from D to L. Wherefore(by the 2 7, propofition)G* L is a parallel line to£> £,forthat the alternate angles D G £and G D K are equal. Wherfore the triangles G D E and LD E are (by this propofition Jeq ual(for they confift vpon one and the lelf fame bafe,namel|y,2)£, and are in the felfc fame parallel lines,namely,£ D and G £)But the triangle LDEis leffe then the triangle £Z>£, Wherfore alfo the triangle GDE is leffe thentfhe triangle F D £.But the triangle G 2) £ is equal to the triangle ABC. Wherfore the triangle ABC is leffe then the triangle D E £; which was requi- red to beproued. But now let the angles B AC and EDF be leffe then two right angles.' and agayne let the angle at the pointed be greater then the angle at the point Z>.Then / fay that the triagle ABCis grea¬ ter then the triangle D E F.Lcttbe fame conftru- ftio be alfo here that was in the two former. And forafmuch as the angles B AC and £ D £,that is, the angles EDG & ED F, arc leffe then two right angle s,but the angles EDG and GD Katz equal to two right angles, takeaway the angle FDG which is common to them both, wherefore the angle remayning, namely, EDF is leffe then the angle remayning,namely,then GD £T:thatis,the angle H D iff which by the 1 f . propofition is e- quall to the angle £ D F) is leffe then the angle G D K. Wherfore the whole angle G D H is leffe then double to the angle G D -KT.But itis double to the angleDG Ffas before it was proued wherfore the angle GDKis grea¬ ter then the angle D G F. PuttheanglcDG Lequall to thcangle GDK (by the 23. propofition Jand produce the line G L till it concurre with the line 8 F alfo produced, & let the concurfe be in the point L. And draw a line from D to b.Andfor as much as the angle DG Lis equall to the angle G D K, and they are alternate angles, therefore the line G Lisa parallel to D £(by the 2 7. propofition ) . Wherefore(by this propofiti¬ on Jthe triangles GDE and L D £ are equal: bu tthe triangle L D E is greater then the triangle FD£,and the triangle G D£ is equall to the triangle ABC. Wherefore the triangle tABCis greater then the triangle D £ £: which was required to be proued, 7 he ift.T'heoreme. T'he fifPropofition* T’riangles which con/r/lvppon equall bafes, and in thefelfe fame parallel lines ,are equall the one to the other . ^^^Fppofe that thefe triangles A BCand V E F Jo con ft ft vpon equal ba* 'W0I- jeSjtbat ts, Vpon B C and E Fyand in thefelfe fame parallel lines ytbat is ^S&'BF and A V.Tben I fay that the triangle ABCis equall to the trian * ofEuclides Elementes . FoL^p* Confirullton^ Demmffraticfi'' gle A B C is equall to the triangle B) E F JProduce (by the fecond petition) the line AT) on echefede to the point es G and H. And (by the 3 i .proportion) by the point B draiPe Pnto CAa paraU lei line B G^and(by the fame) by the $ pointe F dra'Hoe Pnto IDE a parallel lineF 0 Wherfore G BC A and (DEF Hare parallelogrammes.But the parallelograms GBCA is (by the 3 6 proportion) equal to the paralle* logr am me D EF H,f or they conjiH Vpon equall bafesjbat isfB C and E F, and are in the ) elf e fame parallel lines .that is }BF andG H. But (by the $4 .proportion) the triangle A B C is the halfe of the parallelogramme GBC AJor the diameter A B deui • deih it into typo e quail partes'-andthe triangle T) EFisfby tbefame)the halfe of the parallelogramme V EF H^for the diameter FT) deuideth it into typo e* quail partes jBut the h dues ofthinges equall are (by the 7, common fentence)e* quail the one to the other. Wherfore the triangle ABC is equall to the trian* gle D EF. Wherefore triangles Tphicb con fist Pppon equall bafes, and in the jelfe fame parallel lines yare equall the one to the other : TPbich Tpas required to beproued. In this propofition are three cafes. For the bafes ofthe triangles either haue one part common to them both or the bafe ofthe one toucheth the bafe of the other onely in a point: or their bafes are vtterly feuereda funder * And ech of Ecbep bef thefe tales may alfo be diuerfly5as we before haue fenein parallelogrammes con /LfifomfiT {iff mg on equall bafes,and being in the felfe fame parallellines*So that he which diuerjlj . diligently noteth the variety that was there put touching them! may alfo eafely frame thefame varietietoechcafe in this proportion* Wherefore I thinke it nedeles hereto repeat?, the fame agayne-.forhowfoeuer thebafes beput, or the toppessthe manner of confirmation and demonftration here put by Euclide will ferue: namely, to draw parallel lines to the tides. ..'3. Thre cafes m thupropofmon* An addition of Pelitarius* T 0 deuide a triangle geuen into Wo equall partes. Suppofe that the triangle geuen to be deuided in to two equall partes, be A B C .Deuide one of the fides therof, name!y,5Cinto two equall partes fby the 10. propo¬ fition jin the point D. And draw a line from the point D to the point -/Flhe I fay that the two triangles A B D & A C Share Cquahwhich is eafy to proue(by the. 3 8. pro¬ portion) if by the point A we drawe vnto the line B C a parallel line (by the 3 1 .proportion J^which let by H K : for fo the triangles AB D and A D Cs confiffing vppon equal bafes B'D & S)C,and being in the felfe fame paral¬ lel lines H/Cand B C are of neceffitie equall. The felfe Q.i. fame An addition ef I'ehtariut,to tleutde a trian¬ gle into two e- yua/l partes. Nofe, An other addi¬ tion of Peltta- wtes. ConFiruftion. Vemonjlration 7 hefirfl Booty fame thing alfo wil happen if the fide B A be deuided into two equall parts in the point £,and fo be drawen a rightline from the point£,to the point C. Orifthe fide A C be deuided into two equall partes in the point F,andfo be drawen aright line from the point F to the point .5: which is in like manner proued by drawing parallel lines by the pointes B,-and C, to the lines B A and A C, And fo by this you may deuide any mangle into fo many partes as are fig - nified by any number that is euenly euen; as into 14, 16,32. (S^Scc. An other addition ofpelitarius* From any point geuen in one of the fide s of a triangle ,to draw a line Which fiat deuide the trian¬ gle into tWo equall partes. Let the triangle geuen be^CD:andletthepointgeueninthefide BC be A. Itis required from the point A to draw a line which ihal deuide the triangle B CD into two equall partes. Deuide the fide B Cinto two equall partes in the point£. And drawea right line from the point A to the point *D.And(by the 3 1. proposition J by the point E draw vnto the line AD a parallel lineA F: which letcutte the fide D C in the point F, And draw a line from the point ^4 to the point D.Then I fay that the line F deuideth the triangle B C D into two equall partes : namely, the trapefium A B D F is equall to the triangle A C F. For draw aline from E to D,cuttine.the line Ain thepointC. Nowthenit ^ is manifeft f by the 3 8 .propofition) that the two trian¬ gles B E D and C £ D are equall (if we vnderftand aline to be drawen by the point D parallel to the line A Cfor the bafe s B E and E Care equal) .The two triangles alfo DE F and A EF arc f by the 37.propofition)equall:fortheyconfift vponoheand the felfe fame bafe EF, and are in the felfe fame parallel lines A D and E F. Wherefore taking away the triangle EFG which is como to the both,the triangl qAE G fhalbe equall to the triangle DF(/ :wher fore vnto either of the adde the trapefiu CFG £,and the triangle ACF fhalbe equal to the triangle- DEC. But the triangle 'DEC is the halfe part of the whole triangle BCD wherefore the triangle^ C£is the halfe part of the fame triangle B ^D.Wherfore the refidue.namely.the trapefium ABF D is the other halfe ofthe fame triangle. Where¬ fore the line A F deuideth the whole triangle BCD into two equall partes : which was required to be done. Jfi$fThe ipJTheoreme. T he tyfiPropofition. Equall triangles confining upon one and the fame bafe , and on one and the fame fide : are alfo in the felfe fame parallel lines . f. P'ppofe tbstt thefetwo equal! triangles j4.BC andT> BCdo conftjl Vp* ^ pon one and the fame bafe gamely, B C and on one and the fame fide. The I Jay that they are in the felfe fame parallel lines. Vra'We a right line from the point A to the point ID. Noth I Jay that A ID is a parallel line to IB C. For if not , then (by the 3 r. propofition ) by the point A dr all? e Vnto the right line BCa parallel line jEyanddraH? a right line from the point E to the point Folio. of Euclides Elementes. point C. Wherfore) triangle EB Cis (by) n.j propofitio)equal to the triangle A B CJor they conji/l vpon one and the felfe fame bafe, namely fBC^and are in) felfe fame parallels , that is y A E and B C. But the triangle DBCis (by fuppo • fition ) e quail to the triangle ABC, VV her fore the triangle DB Cis equal to the triangle EBCy the greater Vnto the lefle.'tohicb is impofiible. Where • fore the line A E, is not a parallel to the line B CtAnd in like forte may it be proued that no other line befides is a parallel line to B Cohere fore AT) is s a parallel line to B CWherfore equall triangles confifiing vpon one and the fame bafe, and on one and the fame fide, are alfo in the felfe fame parallel linesi'frbicb Tvas required to be proued This propofition is the conuerfe of the 37.propofition.And here is to be noted This Theorem* that if by the point A.you draw vnto the line B C a parallel line, the lame dial of •heconuerfeof neceffitie either light vpo the pointD,orvndcr it, or aboue it. If it light vpo it, then is that manifeft which is required: but ifit light vnder it,then foloweth that abfurdi tie which Euciide puttech, namely, that the greater triangle is equall to the leffe: which felfe fame abfurditiealfo will follow,ifitfall aboue the point D* As for example. Suppofe that thefe equall triangles A*B Cand T> SC do confift vppon one and the felfe fame bafe B C,and on one and the fame fide. Then / fay,that they are in the felfe fame parallel lines.and that a right line ioyning together their toppes is a parallel to the bafe B C.Draw a right line fro A to D.Nowifthisbe not a parallel to the bafe B C,let AS be a parallel vnto it, and let AS fall without the line AD, And produce the line B D till it concurre with the line A E in the pointe E and draw aline from E to C.Whcrfore the triangle tsf B Cis equal to the triangle EB C:but the triangle A*B Cis equall to the triangle D B C: Wherfore the triangle SBC is equall to the triangle D B C. Namely,the whole to the part: which is impofiible. Wherfore the parallel line fal- leth not without the line A D. And Euciide hath proued that it falleth not within. Wherfore the line AD is a. pa¬ rallel vnto the line 3 C. Wherfore equall triangles which are on the felfe fame fide, and on one and the'felfe fame bafe,are alfo in the felfe fame parallel lines: which was required to be proued. An additionofFluffates, The felfe fame alfo followeth in parallelogrames.Forifvpon the bafe AB be Anadditiencf fet on one &c the fame fide thefe equal parallelogrames ABCD sc A B G E,they riujpstes. fhallof neceffitiebem the felfe fame parallel lines.Fori foot, but one of them is O.ii, fet t*. 'J f sirt addition ef Campanula Tbefirfl Hooke feteyther within or without, let the parallelo¬ grams B F being equall to the parallelograme A BCD be fet within the fame parallel lines: wherefore the fame parallelograme B F beyng e- quall to theparallelograme A B C D (by the proportion) (hall alfo be equall to theother pa- rallclograme A B G E (by the firft common fen- tence) For the parallelograme AB GE is by fup$ pofition equall to the parallelogrammeAB C D; whetforc theparallelograme B F being equall to the parallelograme AB G E,the parte fhall bee e- qualto the whole, which is abfurde.The fame in- conuenience alfo will foliowc, if it fall without* Wherefore itcan neither fall within nor withs out.Wherfore equall parallelogrames beyng\rpon one and the felfc fame bafe and on one and the fame fide,are alfo in the felfe fame parallel lines. . / An addition of Campanus. If a right line deride ttyo fide: of a mangle into m>o equall partes : 'it fall be efuidiffant vntd the third fide. Suppofe that there be a triangle ABC: and let there bee a right ly ne D E, which let deuide the two fides AB and B C into two equall partes in the pointes D and E Then I fay ,that the line D E i s a parallel to the line A C4 Drawe thefe two lines AEand D C*Now then imagining a line to bedrawn eby the point E parallel to the line A B, the triangleBDE fhall (by the 38.pro pofition) bee equall to the triangle D A E(for their two bales AD and D Bare putto be equall) And by the fame reafon the triangle B D E is equall to the triangle C E D. V Vherforefby the firft common fen tece) the triangles E AD and E C D are equall,and they are ere&ed on one and the felfe fame bafe,namely ,DE,andon one and the fame fide. VVhereforefby theqp^propofition) they are in the felfc fame parallel lines, and the line which ioyneth together their toppes is a parallel to their bafe. V Vherfore the lynes DE and A C arepa- rallels : which|was required to beproued* be^o.T heoreme. The ^.ofPropofition* Squall triangles confijling vpon equall bafes , and in one and the fame fide: are alfo in the felfe fame parallel lines . iofe that thefe equall triangles A EC and CD E do confift Vppon e* Squall bafes 9 that isy Vppon ft C and CE , and on one and the fame fyde , ti® namely yon the jide of A. Then I fay that they are in the felfe fame paraU lei ofEuclides Elementes. Fol.ji. Ml lines . Draw by the firB peticion a right line from the point A to the point e ’ B)JS[ow I fay that A T) is a parallel line to B E.Forifnotjhen (by the y.propo* Jition)by the point A dray? Vnto the line BE a parallel line A F. And dralbe a right line from the point Fto thepomte EyVherfore(b) the js, propoJitio)the triangle B A Cis equall to the triangle CF E'for they confift Vpon equal bafesy that is B C and C E,and are in the felfe fame parallel lines, namely, B E and A F.But by fuppofttion the triangle A B ^ C is equal to the triangle CDEJ/Vher * fore the triangle DC Eis equall to the triangle FCE, namely s the greater Vnto tbelejfe}'tobich isimpofiible.Wberfore A F is not a parallel line to B Et And in lib forte may Tbe proue that no other line befides A D is a parallell line to B E.Wherfore AD is a parallel lyne to BE Equall triangles therfore confining Vppon equall bafes^and in one and the fame fideiare alfo in the felfe fame pa* rallel lines: lubicb "teas required to be proued. This propofition istheconuerfeot: the38,pcopo.(ition*And inthisas in the former propofition,ifthc parallel linedrawen by the point A, Ihould not paffe by the poincD, it mud paiTe eyther beneath it,or aboue it. Eudide&tceth forth onely the abfurdity which ihould follow if it pafle beneath it: bat the felfe fame abfurditicalfo wdfoliow ifit ihould paileaboue it: as itisnothardto lee by the gatheringthereofinche former propofition* And therefore here I omitte it* The 3 1 . Theoreme. The 4 ifPropoftion . If i par allelograme & a triangle haue one & the felfe fame bafe, and be in the felfe fame parallel lines : the parallel grame/halbe double to the triangle. E T> A let the be in the felfe fame parallel lines, namely fB C&jt E.The I fay ft hat the paralklograme ABCDis double to the triangle B E CJDraief by the firjl peti * cion ) a rygbt line from the point e A to y pointe Cyyherjore(by the 37. propo • Oaf fition Vppofe that the paralklo * grame ABCD and tbetri* angle EBC haue one the fame bafepiamely^ B C, and Ctnftru&M#. Demonfiration leading to an abfurditis^ T hh proportion it the Conner fe of the pofitiiu DemenBfitttem ) (TbejirflcBooJ{C Jition ) the triangle ABC is equall to the triangle EBC-. for they are Vppon one and the ftlfe fame baje & Cy and in the felfe fame parallell lines B C and E A : hut the parallelogram e A B CS) is double to the triangle ABCfby the 34, propofition )for the diameter thereof A Cdeuideth it into tUso equal part slither fore the parrallelogramme ABCS) is double to the triangle E B Clftherfore a parallelogramme and a triangle haue one and the felfe fame bafe , and be in the felfe fame parallels , the parallel gr amefhall be double to the triangle: -frhicb was required to be proued. Tv/o cafet in tbif propofition. This propofition hath two cafes.For thebafebcyng one,the triangle may haue his toppe withoutthc parallelograme, or wichin/The firftcafe is demon- ftrated of the authorfThe fecond cafe is thus. Suppofe that there be a parallelograme AB CD> and let the triangle be E C Neither of which let haue oneand the felfe fame bafe, namely, CDt and let them be in the felfe fame parallel lines CD and A B , and let the toppe of the triangle £ C ^namely, the point £,be within the paralle¬ lograme ABC .D.Then / fay that the parallelograme A 2 C D is double to the triangle EC D. Draw a right line fro the point A to the point D.Now forafmuch as the paral¬ lelograme A B CD is double to the triangle AC D (by the 34-propofition) and the triangle tsfDCis equall to the triangle £ DC (by the 3 7. propofition J.Therfore thepa- rallelogrammc A B CD is double to the triangle £ CD; which was required to be proued. AES jcorolUrj. By this propofition it is manifeft: that if the bafe be doubled, the triangle c* refted vppon it flialbe equall to the parallelogramme. The felfe fame And if the bafes ofthe triangle and of the parallelogramme be equall , the demonfirattm felfe fame demonftration will lerue if you drawc the diameter oi the parallelo- Trifngk & tic grame* F°r the triangles being equal , the parallelogramme which is double to parallelograme the one, fiial alfo be double to the other. And the triangles muft nedcs be equal! beSpon tjHad (by the 38.propofition)for that their bafes arc equal,and for that they are in the felfe fame parallel lines. The conucrfeofthis propofition is thus. If a parallelogramme and a triangle haue one And the felfe fame bafi,or equall bafes the one to t he at her, and be deferi bed on one and the fame fide ofthe bafe : if the parallelogramme be double to the triangle, they Jhalbe in the felfe fame parallel lines. Theconuerfe of For if they be notjthe whole fiial be equal I to his parte. For then the toppe tbit propofition. of the triangle muftnedes fall either within the parallel lines or without. And whether ofEuclides Elementes . F0L52. whether of both foeuer it do, one and the felfe fame impoffibilitie follow, if by the toppe of the triangle be draweti vnto the bafe a parallel line. An other conuerfe of the fame proportion* If a parallelogramme be the double of a triangle, being both Within the felfe fame parallel lines X other cat- then, are thty vpon one and the felfe fame bafe, or vpon equallbafes .F or if 111 that Cafe their has uerfe ofthe ~ fes fliould be vnequal, then might ftraight way be pro ued, that the whole ise- famefr3?cfair>' quail to his part: which isimpoffible* A trapefium hamng two fides onely parallel lines,is ey ther more then don- Compart fort of a ble , or leffe then double to a triangle contayned within the felfe fame parallel triangle and# lines, and hauingone and the felfe fame bafe with the trapefium, or table.! oil the double it cannot be, for then it fhould be a parallelogramme. A trapefium ha? felfe- fame up, uing two fides parallels hath ofnccelTitic the one of them longer then the other: and in the felfe for ft they were equall then fliould the other two fides encloiing them be aifo e- tarallel quail (by the 35*propofition»)Ifthe greater fide of the trapefium be thebafe.of taes* the triangle, then dial th<“ trapefium be leffe then the double ofthe triangle And if the leffe fide of the trapefiu m be the bafe of the triangle then fhall the trapefi- um be greater then the triangle. For fnppofe that AB CD be a trapefium,and let two fides thereof, namely, AB and CT> be parallel lines,and let the fide AB be leffe then the fide CD, & produce the fide AB infinitlye on the fide B to the point i7. And let the triangle E CD haue one and the felfe fame bafe with the trapefium, namely, the line CD.Then I fay that the trapefium AB CD is leffe the the double ofthe triangle £ CjD.Forput the line AF equall to the line C D ( by the 3 .propofitio)anddraw a line from D to F. Wherefore A CD F is a parallelo¬ gramme ( by the 3 3 . propofition) . Wherefore (by the 3 4 propofition)it is double to the triangle £ CD. But the trapefium AB CD is a part of the parallelo¬ gramme A CD F. Wherefore the trapefium AB CD is leffe then the double of the triangle £CD: which was required to beproued. ' Agayne let the triangle haue to his bafe the fide A .S.Then I fay that the trapefium A 2? CD is trea¬ ter then the double of the triangle tAE 2. For from the fide C D cut of the line CF equall to the line A B (by the 3 . propofition) And draw a line from B to F, Wherfore (by the 3 3 .propofition ) <lAB C Fisa. parallelogramme.- andjrherefore is (by the 344*0- pofition J double to the triangle csEEB. Where¬ fore the trapefium B CD is more then thedou. ble ofthe triangle tAEBi which was required to beproued. c r When the grea¬ ter fide of the trapefum is the bafe of the tri- angle. When the leffe fide is the bafe. •. r- 0.iiij, T T3 C&njlruftton. ThefirttHookg The u/Probleme . 'The fyi.propofition. V nto a triangle geuenjo make a parallelngrame equal ftihofe angle [hall bee quail to a reBtline angle geuen. 'Vppofe that the triangle geuen beABCy and let the retliline angle geuen be It is required that Vnto the [triangle ABC there be made a paradelograme equally ‘tohoje angle Jhal be equall to therefliline angle geuen, namely, to the angle DDeuide( by the io ,propojitio)the line B C into two equall partes in the pointe E. And(by the jirH petition) dra'ta a right line from the point A to the point E.And(by the 2$. proportion) Vponthe right Une geuen E C,and to the point in it geuen E}make the angle C EE equal to the angle D.Andfby the 5 1. pro • pojition) by the point A dr alt) Vnto the line E C a parallel line A H: and let the line EFcut the line AH in the point F.and(hy thefame)hy the point C , dralbeyntothe lineFFa parallel line C G.W her fore FECG is a parallelograms. And forajmuche as BE is equall to E Cither fore (by the i proportion) the triangle A . __ BE is equall to the triangle A EC, for they conjM ypo equall hafes that is BE and EC, and are in the felfe fame parallel lines, namely, BCandA H Wherfore the triangle ABC is double to the triangle A E C.And the parade * lograme CEFGis alfo double to the triangle A EC: for they haue one& the Jetfe fame baJeynamely,E Ci and are in the felfe fame parallel lines, that is, EC and AH. Wherfore the parallelograms BE CG is equall to the triangle A B C,and hath the angle C E F equall to the angle geuen D, Wherefore ynto the triangle geuen AB Cis made an equall pqrallelogr ante, namely, F EC Gfyyhofe angle C E Fis equall to the angle geuen D: 'tobicb Tt>as required to he done4 t>emo»pration D - *■> v\\ U 4V.O1 The conuerfe of this former fro- p option. orti n The conuerfe ofthis proportion after Pelitarius . Vnto afarattelogramme geuen, to make a triangle equall, hauyng an angle equall to a reflilint angle geuen. 1 ■ ' Suppofe that the parallelograrae geuen be A B C7), and let the angle geuen be Ea Itis required vnto the parallelogtaine *4 'BC T> to make a triangle equall hauyng an angle ofEucIidesElementes. Fol.%. A ' B m angle equal to the angle £.Vpon the line. CD and to the pointe in it C,defcribe (by the 2 3 . propo- pofition) an angle equall to the angle Ei which let be DCF: aud let the line C F cut the line cA' B being produced, in the point F: and produce the line C I) (which is paralfel’ to the line A F) to the point G.And put the line DGe- quall to the line C D and draw a line from F to Cj, Then / fay that the triangle C F G is equal to the parallelograme ABCD. For for¬ afmuch as(by the 38. propofition ) the whole triangle CF G is double to the triangle CDF . Aud( by the 4 1 .propofition)the parallelograme ABC D isdouble to the fame triangle C D Fi therfore the parallelograme A BCD and the triangle CFG are equall the one to the other: which was required to be done. n The yiftbeoreme. The 4.3. Tropofition . fn euery parallelograme fbe fupplementes ofthoje paralklo* grammes which are about; the diameter ?are efuaUthe one to the other. ? Vppofe that ABC (Dbea parallelograme, and let the diameter then of be A C: and abou t the diameter A Clet thefeparadelogrames EM and O Fconfift: and let the fupplementes be B Kjmd KJD. Then I fay that the/upplement B Kjs equall to the fupplement K,!D. For fora/much as ABCD is a parallelograme and the diameter therofis A C} therfore & ( by the 3 4. propofition ) the triangle A B C is equall to the triangle A£>C, A * gayne forafmuch as A EKJA is a pa* ratlelograme3and the diameter therofis A therfore (by the fame) the trian* gle A E Kjs equall to the triangle A M K.And by the fame reafon alfo the tri¬ angle KJFC is equall to the triangle K^ G C, And forafmuch as the triangle A E Kjs equall to the triangle A HK+and the triangle i^FC to the triangle K G C, therfore the triangles A E Kjtnd K, G C are equall to the triangles A H K and K.EC: and the whole triangle A B Cis equall to the whole triangle A !D <Pf C v<- Thefir/i^ooke H o tp parallclo- grammes are fajdeto conffle about a dia¬ meter. C: Tbherfore the refidue, namely ,the fupplement © KJs (by the common fen* tence) e quail to the refiduejiamelyjo the fupplement I(J) . Wherefore in e* uery parallelogr amme , thefupplementes of tbofe parallelogrammes 'tobicbe are about the diameter, are equall the one to the other : Tbhiche Tt>as required to be proued , Thofe paralldcgrames are fayde to confift about a diameter which haue to their diameters partofthc diameter ofthe whole and great parallelograme, as in the example of Euclidc, Andfuchparallclb- grames whichhaue not to their diameters part ofthe diameter ofthe greater parallelograme, are fay de not to confift about the diameter,^ or . the the diameter ofthe greater parallelograme cutteth the fide outlie Idle cocayned wy thin it. As in the parallelogramme A i?,thc diameter <7Z>, cutteth the fide EH of the parallelogramme CE. Wherefore the parallelogramme CEis not about j one and the felfe fame diameter with the parallelo- grammeC®. > H C r~ ■ yAf: y . t • ; > * E . , 3 Supplementes or Complementes are thofe figures, which with the two pa- Sctmpkm!nS. lallelogrpmmcs accomplifinhewholeparan-clogramme. Although Peiitarius for diftmttion fakeputteth a difference betwene Supplementes and Comple¬ mented, The parallelogram mes aboutthc diameter he 'calieth Complementes, the other two figures he calieth Supplementes* Three cafes in thisTheorcmc. 7 he firft cafe. ... This theorerae hath three cafes onely* Fortheparallelogrammes which confift about the diameter, eythcr touch the one the other in a point: or by accc ray ne parte of the diameter are feuered the one from the other: or els they cutte the one the other . For the firft cafe is the example ofEuclidebefore fct. The fc- cond cafe is thus. • ' • : ’ The fecond cafe. Suppofe that AB be a parallelograme? whofe diameter let be CD : andabpute tfie fame diameter let thefe parallelogra’mmes C KandD L confift: and betwene the let there, be acertavne part of the diameter, namely, . L K.Then I fay that the two fupplementcs A O L ICE & IS F KL //are equall. For we may as beforefby the 3 4.propofition jproue that the triangle AC D, is eq uall to the triangle B CD, and the triangle E C K to the triangle K CF, and alfo the triangle D G L to the tri- c angle D H Lt Wherfore the refidue, namely, the fiuefided figure AG L KE is equall to the refidue, namely, to the flue fided figure B F K L Hi that is, the one fupplement to the other .which was required to be proued. The th>rd cafe. But now fuppofe &A B to be a paralleIogramme,and let the diameter thereof be CD: and let the one of the parallelogrammes about it be E C F L9 and let the other be s? :* DGKHi of Euclides Elementes. FoL 54. A Jp DGKH,o£which let the one cut the other. Then I fay that the fupplementes F G and EH areequall.For forafmuchas the whole trian¬ gle DGK is equal to the whole triangle DHK fbythe jq.propofition J, and part alfo of the onemarndy, the triangle KL Mis equal to partoftheother.namely^to the triangle KL iV(by the fame), forLKis aparallelograme i therefore the refidue,naraelyJthe Trapefium ©TA^Ais equall to the refidue, namely, to the trapefiu D L MG* but the triangle ADC is eq ual to the triangle BCD, and in the pa- railelograme E F, the triangle F CL is equal! to the triangle £C£,and the trapefium D G MLisf asit hath bene proued) equal! to the trapefium DHNE. Wherefore the refid ue, namely, the quadrilater figure G F is equall to the refidueuiamely, to the quadrilater figure E Hjhxt is, the one fupplement to the other:which was required to be proued. This is to be noted that in ech ofthofe three cafes it may fo happen, that the parallelogrammes aboute the diameter {hall not haue one angle common wy th the whole parallelogramme,as they haue, in the former figures.But yet though they haue not, the felfe fame demonftration wil feme, as it is play ne to fee in the figures here vndcrneathput*Foralwayes, if from thinges equall be taken a^ay thinges equall,thc refidue fhalbe equall. / This propofition P elitarius calleth Gnomicall, and mifticall,for that of it (faythhe) fpring infinite dcmonftrations,andvfes in geometry. And he putteth the conuerle thereof after this manner* andmifttcal. V ; * " - ’ '( i < ' * * 'V If a parallelogramme be derided into Wo equall fupplementes ,and into Wo complements What- The conuerfe of foeuer: the diameter of the Wo complementes JhaH be fit direttly , and make one diameter of the this propofition. Vehole parallelogramme. Here is to be noted as 1 before . .0 lied thatpeli tarius for diftindion fake putteth a diflerencebetwene fupplementes and complementes /which diffe- rencejfor that I haue before declared, I fhallnot needeheretorepeteagayne. Suppofe that' there be a parallelogramme A BCD, whofe two equall fupplements let be A £ F (./ and F HD K,and let the two complementes thereof be GF C K and £5 ££T: whofe diameters let be CF and FA.Then I fay that CFB is one right line, and is thediameter of the whole parallelogram me A B CD\ forifit be not, then is there an F-ij* other / GtnRruttisn. Tbefirft'Bookp other diameter of the whole parrallelogrammewhich let be C L B being drawen vnder the diameters C F and F B, and cutting the the in the point L,And( by the 1 1 .propolition ) by the point L,draw vnto the line a A C a parallel line M L N. And fo are there in the whole paral¬ lelogramme A'B CD two fupplements AMGL and L H N D, which by this propolition lhalbeequall the one to the other. For that they are about the diameter C L B. Butthefupplementex4r£.F(7is fbyfuppolition) equall to the fupplcment F H D K:and forafmuch as FHD Kis greater then LHDN,AEFG alfo dial be greater then A MGL,nameIy,the part greater then the whole: which is impolfible. And by the fame reafo may it be proued,that the diameter cannot be drawen aboue the diameters C F -t and F B. Wherefore C F B is one diameter of the whole parallelogramme tABCD: which was required to be proued. H*The rz. Trobleme . The 44. Tropofition . Vppon a right line geuen Jo apply e a paraUelograme equall to a triangle geuen, and contayningan angle equall to a rev * tiline angle geuenm 7M V pp°ft that the right litiegeuen he A !E,and let the triangle geuen yjibe Cyand let the reSliline angle geuen be (D.Itis required Vpon the I right line geuen AByto apply e a parallelogramme equal to the trian* gle geuen Cy and contayning an angle equall to therectiline angle ge* ue (D,T)efcribe(by t he 4 fypropo/i tion)y paraUelograme EG EF equall to the tri* angle Cy and hailing the angle <BGFe* quaU to the angle (D.And vnto the line E <B ioyne the line A E in fuch fort that they make both one right line. Jnd extend the line EG beyond the point G to thepoynte HtAnd(by the ]ipropo/ition)by the point A draive to either ofthefe lines E G and EE a paraUellme A H. And (by the fir ft peticion)draf» a right line from the point Hto the point E.And forafmuch as vpon the parallel lines A Hand E Efalleth a certayne right line H F, there fore(by the 29 propofition)the angles A HEandHEE>are equaUto two rightangles: therefore the angles E H G andGFE are lejje then th>o right angles : but if ypon tivo right lines fall a right line making the inward angles on one and the fame I ofEuclides Ekmentes. FoL $£* fame fide lejfe then tfro right angle s^thofi right Itnes being infinitly produced JhaU at the length mete on that fide in which are the angles lejfe then tfro right angtes(by the 5, petition). VVherfore the lines HBand FE being infinitly produced "frill at the length mete, Let them be produced^ let them mete in the point K^And (by the 3 1 propofition) by the point KJlrafr to either of thefe lines EA and F Ha parallel line K^LfAnd (by the 2. petition) extend the lines H A and GB till they cocurre "frith the line H\L in the pointes L and M. Where* Dem»njtr*tio» fore H L EJFis a parallelogramme yand the diameter thereof is H Kg and a» bout the diameter HKjre the parallelogrammes A G and M Ey and the flip* plementes areLB and BFifrhere fore (by the 4 3 , propofition) the (up plem en t LB is equall to thefupplement BFibut by confiruttton the parallelograme BF is e quail to the triangle Cifrherefore aljo the parallelogramme L Bis e quail to the triangle C,And forafmuch as the line F His a parallel to the line Ly and Vpon them lighteth the line G Mother e fore (by the 27. propofition) the angle FG Bis equall to the angle B ML, But the angle FG B is equal 1 to the angle !Dytherfore the angle B ML is equal to the angle D. Wherfore Vpo the right linegeuen A B is applied the par rallelogr ante L B, equal to the triangle geuen Qand contayning the angle B ML equal to the reFlilme angle geuen iD: frhicb fras required to be done* Applications of Applications oflpaces or figures to lines with exceffes or wantes is (fayth ^ff soffalTs Eudemus) an auncient muention of Pithagoras. a»au»dent inn Mention of Pi- When the fpace or figure is ioyned to the whole line, the is the figure fayd it to be applied to thcline.But if the lengthof the fpace be longer then the line, the fajde to be ap- it is fay de to exceede: and ifthe length of the figure be fhorter then the hne,fo r^dtoaiine. that part of the line remay neth without the figure deferibed, then is it fayde to vant. In this probleme arc three thinges geuen*A right line to which the applica- ffZfJthss tion is made, which here muftbe the onefideofthe parallelogramme applied, A propofition , triangle whereunto the parallelogramme applied muft bee equall : and an angle wheruntothe angle of the parallelograme applied mull: be equally And if the an¬ gle geuen be a rightangle,the fhal the parallelograme applied be either a fquarc9 or a figure on the one fide longer* But if the angle geuen bean obtufc or an acute angle9then flia.ll the parallelograme appliedbe a Rhombus or diamond fi- gurejor els a Rhomboides or diamondlike figure* The Conuerfe of this propofition after Peli tarius* Vpon a right line geuen ,to applie vnto a parallelograme geuen an equall triangle hauyng an an* gle equall to an angle geuen. T^* conuerfe of tba propofition. Suppofe that the right line geuen be aA J?,and let theparallelograme geuen be C D B F*and let the angle geuen be G. It is required vpon the line A B to deferibe a tria- gleequall to the parallelograme CD E F, hauing an angle equall to the angle G. Drawe the diameter CF&produceCD beyond the pointD to the point H. And put the line P.iij, D H ThefirflUoohg Cong/uBion„ D£/ equall to the lin eCD. And draw a line from F to H. Now the (by the qi.prqpofition) thetria- gle C H F is equall to the paralle- lograine CD E F ^nd(by this propofition)vppon the line B defcribe a parallelograme ABKL equall to the triangle CHi^ha- uing the angle AUL equal to the angle geuen G; and produce the line B L beyonde the pointe L to the point M.And put the line LM equall to the line BL,and draw aline from A to Mv Then I fay-that vpon the line AB is defcribed the triangle A B M,which is fuch a trian¬ gle as is required. For (by the 41 .propofition)the triangle A BM is equal to the paral¬ lelogramme AB K L( for that they are betwene two parallel lines B M and A K, & the bafe of the triangle is double to the bafe of the parallelogramme) :but A 3 K L is by conftrudion equall to the triangle C H F:and the triangle CHFis equall to the pa¬ rallelograme CD E F.Wherfpre (by the firil.cpmmon fentence) the triangle ABM. is equall to the parallelograme geuen CD£ isandhath his angle A BM equal to the an¬ gle geuen C: which was required to be done. *'■ vv - . i * • s v? • *, The ifTrobleme, The ^Tropofition* r ‘ ' 1 u) ri ‘ ' • 'A ‘ ' l Ky r. j v . To defcribe a parallelograme equal to any reBiline figure ge* uen^andcontayning an angle equall to a reUiline angle gene. i. L’a Demon fir at ion' Vppofe that the re Biline figure geuen he 3 C(D,and let the reBilim angle getie he E,It isrequired to de/crihe a parallelograme equall to the reBilme figure geuen SiftCD^and contayning an angle equal to the re* Bilim angle geuen E.(Dra'to(by thefirfi: peticioma right line fro the point 0 to the point n.And (by the y2,propofition)vnto the triangle Aft l D defcribe an e* quail parallelograme F Mfiauing his angle Fjf^FI equall to the angle E. And (by the 4dtofthefirfi) bpo the right line G H apply the parallelogramme G A i equal to the triangle VBCfia* uing his angle GUM equall to the angle E. And for aj much as eyther of thofe angles H EfF and G H M is equall to the angle E : therefore the angle MEfFis equall to the angle G TIM: put the angle EfM G com * mon to them both fibber fore the an * oles F KJFd and KJAG are equall to the angles EfH. G and GUM. but the angles F E^M and EfMG are (by the ~g,propofition) equall to tn?Q right angles yEherf ore the angles KJA G and G MM are equall to two right ofEudtdes Elementes. FoL*>6 . right angles.lSlow then Vnto a right line G H,and to a point in the fame H^are drathen two right lines Kjtdand HM not both on one and the famejidey ma <* king the fide angles equall to ttvo right angles, W her fore (by the i^propofiti* on) the lines K^H and HM make dire Elly one right line. And forajnmch asV* pon the parallel lines J\M and F Gfalleth the right line H Gy therefore the alternate aagles M HG and H G Fare (by the 2<y propofition fequali the one to the other : put the angle H G L common to them both^Vherfore the angles MHG and H G Fare equall to the angles HG F andHG L 'But the angles MHGejr HGL are equall to ttvo right angles (by ey 29^ propofition), VF her* fore alfo the angles H G Fand HG L ire equall to tlvo right angles. VVber* fore (by the \4,propofitiori) the lines FG and G L make dire Elly one right line. And f ora/much as the line E\Fis (by the 24.fr opofition) equal to the lyne H Gs and ft is alfo parallel Vnto it: and the line HGisfby the fame) equall to the line MLjtherfore (by the first common fentence) the line F Kjs equall to the lyne MLyand alfo a parallel Vnto it (by the 3 o,propofition), But the right lynes M and F L ioyne them together yVherf ore (by the $ $ ,propo(ition) the lines E( M andF L are equall the on to the other and parallel Itnes.VVberfore IfiFLM is a parallelograme.And forafmuch as the triangle A BID is equal to the paraU lelogrameF H}and the triangle DBCto the parallelogramme G M ; therfore the whole reEliline figure ABCDis equall to the tvhole parallelograme Kfih M. VF her fore to the reEliline figure geuen ABCDis made an equall parallel grame KJE LM tyhoje angle F KJM is equal to the angle geuen jaamelyfo E: iv fitch 'tv as required to be done. The rc&ilinc figure geue is in the example of Euclide is aparallelograme.But if the re&ilinc figure be of many fidcs,as of 5.5. or mo,themuftyo« refoluc the figure into his triangles,as hath bene before taught in the 3 2 ♦ propofition. And the apply a parallelograme equal to eucry triangle vpon a linegcue.as before in the example of the author. And the fame kind of reafoningwil feme that was be fore, only by reafoofthe multitude oftriangles,you ihall haue neede ofokener repeticio ofchez^.and i4*propofitiostoprouethatthebafes ofal the parallel o« grames made equall to all the triangles make one right line, and (c alfo of the toppes ofthefaidparallclogramesJPelitarius addeth vnto this propofition this Pro'oleme following* T vne quail reEliline fuperficieces beyng geuen ,to find out the excejfe of the greater aboue the lejfe. Suppofe that there be two vne- quall re£filine fu- perficieces A8cB ofwhichlet^be thegreater. It is required to finde out the excefle of the (uperficies A aboue the fuper- ficieces B . De- feribefby the 44. P.iiij. pro- addition of PelitarsM. 'ThefifUHooke v propofition)the parallelograme CDE F equall to the rediline figure -^,contayning a right angle. And produce the line CD beyond the point 'Z> to the point (j : & put the line T> Cj equall to the line C D. Andagaine (by the 44.propofition ) vpon the line D G dcfcribe the parallelograme DG H X equall to the rediline figure “2?, and hauyng the angle D G Ka right angle. And produce the line K H beyond the point H,vntiliit cutte the.Iine C £ in the point L.Then I fay that H L E.F, is the excefie of the rediline figure c/Zaboue the rediline figure2?,Fornrft. that CGKL isaparallelogrammeit is mani- feft, neither nedeth it to be demonftrated. And forafmuch as the lines C'D and *D G are by fuppofition equal and either of them isaparallel to K L,therfore(by the 3 G propo- fitionjthe two parallelogrames CH and D /fare equall. And forafmuch as D K is fup- pofed to be equall to the rediline figure £,C//alfolh all bo equall to the fame rediline figure rB. Wherfore forafmuch as the whole parallelograme CFis equali to the rediline figure a^andL F is the exccffe of CF aboue D L or D K,it followeth that L F is the ex- celfe of the rediline figure aboue the rediline figured : whiche was required to be done. An other more redy 'fray, i Let the parallelograme C DEF remayne equall to the rediline figure A, & produce the line C £> beyond the point d to the pointe G . And vpon the line D G deferibe the parallelogrameDG HKequall to the rediline figure B. And produce the lines EC & H K beyond the points C and K till they concurre in thepoiut L. And by the pointe D draw the diame¬ ter LDM, which let cutte the line HGbcyng pro¬ duced beyonde c the pointe G in ^ the point M, & by the pointe M drawc vnto the E L ] sc H . . . D - - - • - - » " • - - - ? - V J ' A ' J M lincHLaparal- * fel M N cuttyng the line E L in the pointe N : and by that meanes is H L M N a paralle* lograme.Then / fay that N F is the eveefie of the rediline figure aboue the rediline figure £.For forafmuch as the parallelograme H D is equall to the rediline figure /?, & the fupplementes H D and D N ar (by the 43 .propofition ) equall : therfore D N alfo is equall to the rediline figure B,which rediline figure D N being taken away fro the parallelograme C F (which is fuppofedto be equall to the rediline figure A) the refi- due N F mall be the ezcelfe of the rediline, figure A aboue the rediline figure £: which was required to be done. The \\fProbleme \ The 45. Tropojition . Vppon a right line geuen Jo deferibe a fquare. 3 ^at right linegeuen be A BJt is required Vpo the right * line A Bjo deferibe a fquare. Vpon the right line A (8, and from a point in itgeuenynamely y A9rayfe Vp (by the n propojitton) a per # pen diculer line A C. And (by the $ propofition) Vnto A B put an e* quallline AT). And (by they.propojitm) by the point T) dratve Vnto AT a parallel line D E. And (by the fame) by the point B dr awe Vnto A V a parallel line Fol.ij. : ; siolm: . iH, a ! ; t> f h::n r:- ■ 5S2SJ £ of Euclides Elementes • line SB, Wherefore AD E B is a pa* raUelogrammeVVberefore the line A 3$ is e quail to the line D Ey and the line A D to the line B E: but the line ABise* quail to the line A Dwherefore thefefdtP "User lines B AyA DfD E,E B, are equall the one to jthe other. Vlfhsrefore the pa* rallelogramme ADEBconpfteth of e* quail fides . I fay alfo that it is re&angle. For forafmuch as Vpon the parallel lines A B and DEfa Ueth a right line AD:thtre * foref by the i ypropofition) the angle. : B .AD and A t) Eare equal to two right anglesihut the angle BA 3) is a right angkyV her fore the angle AD E alfo is a right angle , But in parallelogrames the fides and angles which are oppojite are e quail the one to the other(by the $ 4 proportion). Wherefore the two oppojite angles A B Eand BED are ech of them a right angle. Where fore the parallelograme A B ED is reBangleiejr it is alfo pr oued that it is equilater . VVherfore it is a (quart it is defer ibed Vpa on the right tmegemnA B\ which Was required to be done # 3 tv ; 'y.\ i. £ ftration-i ere&ed the perpendicular line CA vpon the line A 2?,and ,put the line A E equall to the line A B : then open your compalfeto the wydth of the line ABorAE,tk fetone foote thcreofin the point £,and deferibe a peece of the circumference of a circle: and againe make the centre the point 2?,and deferibe alfo a piece of the circumference of a circle-jUamelydn fuch fort that the peece of the circum- ferece of the one may cut the peece of the circumference of the other,as in the point D : and from the point of the interfei5tion,draw vnto the points E & B right lines: Sc fo ihaibe deferibed a fquare,As in this figure here put,wher- a ^ in / haue not drawen the lines E D and Z> £,that the pee- ** ces of the circumference cutting the one the other might the plainlier be fene. To defer i be * Mire mecket- ntcuUj, An addition ofProclus. If the lines vpon Vchich the fquares Suppofe that thefe right lines A B and CD be equall, &vpon the line AB deferibe a fquare A BEG :and vpon the line CD de¬ feribe a fquare CD HF. Then I fay that the two fquares ABE GScCDHF are equal* For draw thefe right lines C2?and HD. And forasmuch as the right Iines./45andCDare equall, & thelinesoA G and HC are alfo equall,and they contayne eqaul be deferibed be equall, the /quotes alfo are equall. angles The eo'tuerfe fhertf* Conflruttiott, Thefir/ffiooke aogles,i|amely,right angles (by the definition of a fqiiafe)thercforefby the 4. ptopo* fition Jthe bafe B G is equall to the bafe//2).And the triangle ABG is equal! tothe triangle C D //.Wherefore the doubles of the faide ttiaiiglcs are equall. Wherefore the fquare A E is cquall to the fquare CFi which was requited to be proued. ■■■ . : ' • ' 4 . ■ . ■. n ns 5 ’H-, 1 o;i v'.. The conuerfc thereof is thus*. \ If the fquares be equall: the lines alfo vppon Vphicb they are defeated are equall. Suppofe that there be two cquall fquares A F and C Cdefbribed vpon the lines A B Sc B (f. The I fay,that the lines A B andP C are cquall.Put the line *4£dire&ly to the line 2?C,that they both make on right line, And forafmuch as the angles are right angles,ther- fore alfo(by the i4.propofition)the right line FB is fet dire&Iy to the right line B G • Dr awe thefe right lines FC^ AG,AF, and C G .Now for afmuch as the fquare AF is cq ual to the fquare CC,the triangle alfo A F B (halbe equall to the' triangle CB G: put the triangle BCF comon to them both. Wherfore the whole triangle A C ( Pis equall to the whole triangle C F G.Where- fore the line AG is a parallel vnto the line CP (by the 38.propofiti6)-.for the triangles confifl vpon one and the felfe fame bafe, namely C P. Againe forafmuch as either of thefe angles AF G Sc CBG is the halfe ofaright angle, therfore { by the 1 7 .propofition jthe line AF is a paral¬ lel to the line C G. Wherfore the right line AF is equal to the right line C G( for the op- pofite fides of a parafielograme are equal! ). And forafmuch as there are two triangles A'BF and B C^.whofe alternate angles are equali,namely, the angle A FB to the an¬ gle BGC, and the angle B AFto the angle B C G’,and one fide of the One is equall to one fide of the other,namely,the fide which lieth betweue the equal angles,that is, the fide A F to the fide CG, therefore (by the 26. proposition) the fide A Bis equal to the fide B C,andthe fide B F to the fide B G. Wherefore it is proued that the fquares of the finest P andCC being equalhtheir fides alfo llialbe equall: which was required to be proued. he 33. Theoreme* ^The 4.7. 'Profofition. In rectangle triangles y the fquare whiche is made of the fide that fubtendeth the right angle Js equal to the fquares which are made of the fides containing the right angle. ^'Vppofe that ABC be a reSlangle triangle, hauyng the angle BAC a right angle. The I fay y the fquare Tubieh is made of the line BC is equall to the fquare s'fthkb are ]made of y lines AB and JiC. Defcribe(by y ^6.propoJti cion ) Vpotiy line BCa fquare BBCE^and (byy fame) Vpon the lines 'BA and AC deferibe the fquares ABEG and AC f^H.Jnd by the point A draH> ( by the pro * 'pofttion ) to either of thefe lynes BID and CE a parallel line ofEuclides Elements s. F0/.58. line A LtAnd (by the firEt petition) draw a right If ne from the point A to th e point B)yand an other from the point C to the pomt KAnd fora /much as the an* gles B A Cand B A G are right angles ,therf ore Vntoa right line B Ay and to a point initgeuen Ay aredrawen two right lines A C and AGynot both on one and the fame fide , makyng the two fide angles equall to two right angler. wher for e(by the 14. propofi * tion)the lines A C and A G make di* re Elly one right line. And hy the fame reafon the lines B A and A H make alfo dire Elly one right line , And fora afmtich as the angle B)BC is equall to the angle FBA ( for either of the is a right angle)put the angle A B C common to them both : wherfore the "whole angle D BA is equal! to the ' whole angle F B CAnd forafmuch as thefe two lines A B and B B) are equal to thefe two lines B F and B C,the one to the other ,and the angle t>B A is equal to the angle FBC: therfore(by the ^.propofitionfihe bafe A B) is equall to the bafe F Cyand the triangle ABB) is equal! to the triangle FB C, But (by the ji. lellynes, that isyBB)and A Land (by the fame) the / quart GBit double to the triangle FBC, for they haue both one and the felfe fame bafe , that is y B F, and are in the felfe fame parade Hynes , that is, FB and G C. But the dou* hies oft hinges equal f are (bythejixte common f entente) equall the one to the other. Wher fore the parallelograme B L is equall to the fquare G B% And in like forte if (by the firft peticion ) there be drawen a right line from the point A to the point E^and an other from the point B to the point We may prone y the parallelograme CL is equal to the fquare HCyVher fore the whole fquare BB) EC is equall to the two fquares G B andHC.But the fquare BP EC is defcribed Vpon the line B Cymd the fquares G B and HC are defcribedvppon the lines BA^rAC: wher fore the fquare ofthefideBCjs equal to the fquares of thefidesBAand A CyVhereforein reElangle triangles, the fquare whiche is made of the fide that fubtendeth the right angle js equal to the fquares which are made of the j ides contayning the right angle: which was required to be de* monfirated. This moft excellent and notable Theoremc was firft inacntedof the create philofopher Pithagorasawho for the exceeding icy concerned of the indention therofjofferedin facrificcan Oxeaas recorde Hierone, Proclus, Lycius,& Vi- trmmis.And it hath bene commoly called of barbarous writers of th" latter time Dulcarnon. Q.ji. An I’ithagoras the firft snuenter of this propofttion. An addition of Pektarius. An other adit to of Pelstarius. Another additi on of -Pelt farms. Thefir/lTloofy An addition of F elitarius. To reduce two vncquall fquares to two equall fquares* Suppofe that the fquares of the lines tA B and./! C be vnequall.lt is required to re¬ duce them totvoequallfquares./oynethetwolinese^Sand^Cat theirendes in fuch fort th at they make a right angle B A C.And draw a line from B to C. Then vppon the two endes B and C make two angles eche of which may be equal to halfe a right an¬ gle (This is done byereftingvpon the line 3 Cperpe- diculer lines.from the pointes B and C : and fo (by the p. propofition ) deuiding eche of the right angles into two equall partes) : and let the angles 3 CD and CBD beeither of the halfe of a right angle. And let the lines B D&ndCD concurre in the point D. Then/fav that © the angle at the pointe D is (by the 3 2. propofition) a right angle. Wherefore the fquare of the fide 3 Cise- qual to the fquares of the two fides D B and D C (by the47. propofition) :butitis alfo equall to the fquares of the two fides A Bund A C(by the felffame propofition) wher- forer by the common fentencejthe fquares of the two fides B D and D Care equall to the.fquares of the two fides ABztxdA C ; which was required to be done. An otheradditionofPelitarius* If mo right angled triangles haue equall bafes. the fquares of the tVeo fides of the one are equall to the fquares oft he ttyo fides of the other. 1 his is manifeft by the former conftmftionand demonftration* An other addition of pelirarius. Two vnequall lines beinggeuen, to knoty hoW much the fquare of the one is greater then the fquare of the other , Suppofe that there be two vnequal lines AB and B C.-of which let A3 be the grea¬ test is required to fearch out how much the fquare of A B excedeth the fquare ot'3 C.Thatis I wil finde out the fquare, which with the fquare of the line B C lhalbe equal tothefquare ofthelinev! .8. Put the finest Aand^Cdircftly,that they make both one righ t line.-and making the centre the point B, and the fpace BA deferibe a circle ADE,&txd produce the line tAC to the circumference, and let it concurre with it in the point f’.And vpon the lyne A E and fro the point C ere& (bv the 1 1. propofition) aperpendiculerline CD, which produce till it concurre with the circumference in the point!): & drawaline from B to Z>.Then I fay,that the fquare of the line CD, is the excefle of the fquare of the line A B aboue the fquare of the line B C. For for- afmuch as in the triangle 8CjD,theangle at the point C is a right angle, the fquare of the bafeZ?D is equall to the fquares of the two fides B Cand C D(by this 47. propofition).Wherefore alfo the fquare of the line A B is equall to thefelfe fame fquares of the lines rBC and CD. Wherefore the fquare of the line B C is fo much leflfe then the fquare of the line A B, as is the fquare of the line C D; which was required to fearch out, Aq the two fquares of the fides B D and C D, are equall to the two fquares of the fides A B and A C.For (by the 6 propofition)the two fides and D Carceauall and Fol^p. ofEuclides Elements* . An other additionofPelitadus. T he diameter of a fij Hare bcinggeuenjtogcM the fquare thereof An other aditio of Pehtarius, This is eafie to be done. For ifvpon the two endes of the line be drawen two halfe right angles,and fo be made perfed the triangle then flialbe defcnbed half of the fquarejthc other halfe whereof alfo is after the fame manner eafie to be de? fcribed. Hereby it is manifejfthat the fquare of the diameter is double to that Square Vphofe diameter it is. ^ Corroltarj The i\.Theoreme. The 4.8. Tropofition \ If the fquare which is made of one of the fides of a triangle? he equal! to the fquare s which are made of the two other fides of the fame triangle: the angle comprehended vnder thofe two other fides is a right angle . “ Vppofe that ABC be a triangle, and let the fquare "tobich is made of one ! of the fides there jiamely yof the fide B Cfe squall to the fquares which are made of the fides BA and A C.Then I fay that t he angle B AC is a right angle. 3(yyfe’vp(by the u.propofitio) from the point Avnto the right line A C a perpendicular line A V, And (by the thirde propofition) Vnto the line A B put an equall line A 3), And by the firfl petition draSo a right line from the point V to the point C. And for af much as the line 3) A is equall to the line A B« the fquare Tbhich is made of the line 3) A is e* quail to the fquare Srhiche is made of the line A B But the fquare of the line A C, common to them both . Wherefore the fquares of the lines 3) A and AC are equal to the Jquares of the lines BA and AC, , But (by the propofition going before) the fquare of the line 3)C is equal toy fquares of the Imes.AV and A C. (For the angle VAC is a right angle) andthe fquare of B C is (by fupp option ) equall to the fquares of A Band AC, Wherefore the fquare of 3) C is equal! to the fquare ofB C therefore the fide V Cis equall to the fide BC, And (ora/much as A B is equall to A V and AC is common to them loth, therefore the fe two fides V A and AC are equall to thefe t^o fides BA and A C, the one to the other, and the hafe V C is equall to the hafe B C\St>her - fore (by the propofition) the angle 1) ACis equall to the angle B A C But the angle VAC is a right angle therefore atfo the angle B AC is a right angle. If Q.dtj. there * This propofition it the Conner (e of the former. *in other De- monflration af¬ ter Pelt tart us. cThefirHcBooke therefore the fquare Tbhich is made of me of the fides of a triangle^ he equal 1 to the fquaresfbhicb are made of the ftoo other fides of the fame triangle , the an * gle comprehendedvnder thoJetTro other Jide sis a right angle : "tohich was re* qniredtohe proued. - This propofition is the conuerfeofthe former,* d is of Pelitarius demon* ftrated by an argument leading to an impoffibilitie after this maner* S uppGfe that ABC be a triangle : & let the fquare of the fide AC, be equal to the fquare s of the two fides A B and B C.Then/ fay that the angle at the point ^which is oppofite to the fide AC yis a right angle. For if the angle at the point c B be not a right angle, then fhal it be eyther greater or lefle the a right angle. Firft let it be is greater. And let the angle DBC be a right angle, by erecfing from the point B a per¬ pendicular line vnto the line BC( by the n, propofition) which let be B D: and put the line #£> equall to the lyne A B (by the thirde propofition And drawe a line from Q to D. Now (by the former propofition) thefquare of the fide CDfhalbe equall tothefquares of the two fides BD a nd# C : wherefore alfo to the fquares of the two fides B yjfand BC. Wherefore the bafe CD fiialbe equall to the bafe C A, when as their fquares are equall : which is con¬ trary to the 24.propofition.Forforafmuchasthe angled BCis greater then the angle DBC, and the two fides A B and BC are equall to the two fides D B and B C, the one to the other, the bafe C A (hall be greater then the bafe CD, It is alfo contrary to the 7.propofition, for from the two endes ofone & the fame line, namely, fro the points B & Cihould be drawn on one and the fame fide two lines B D and D C ending at the pointe D, e- quall to two other lines BA and A Cdrawen From the fame endes and endin g at an other point, namely,at A, which is impof- fible.By the fame reafon alfo may we proue that the whole angle at the pointe B is not lefle then a right angle. Wherfore it is a right angle: which was required to be proued. (’••) The endc of the firft hoo^e ofSucl'tdes Slementes , €j The fecond booke of Eu- 60 * elides Elementes* N this fecond booke fidclide ilie^etK , t-ha^ is a Gnom6,anda right angled parallelogramme. Alfo in this bobke are let forth the powers cilines,deui- ded euenly and vneuenly , arid of lines added one to an other. Thepower ofaline , is the fquare of the fame line: tffttis, afqitat^ euery fide of which is e- quail to tire line* So that here are fct forth the quali¬ ties and proprieties of the fquarcs and right lined fi- guresjwntch are made oilines & oftheir parts. The Arithmetician alfo out ol this booke gathered! ma? ny compendious rules of reckoning, and many rules alfo of Algebra, with the equatios therein vied. The groundes alfo ofthoie rules are for the m oft part by this fecond booke dernon- ffrated* This booke moreouer contayncth two wondbrfull propoficions* one of an qbtufe angled triangle, andthe other ofan acuterwhich with the ayde of the 47*propofition ofthefirft booke ofEuclide, which isofa re&angle triangle,ol how great force and profite they are in matters ofaftronomy,they knowe which haue trauayled in that arte* V Vherefore if this booke had none other profite be fide* onely forthcfez*prapofitions fake it were diligently to be cmbracedand ihidied. The argument of the Jecond books • fVhat it the poTver of a tine . Manycompe * dious rules of reckoning ga- theredout of this bookstand aljo many rules of Alge¬ bra* Ttpo Wonder* full proporti¬ ons in this ejimtions . i. Suery reBangled parallelogramme, is [ayde to be contayned vnder two right tines comprehending a right angle* A parallelogramme is afigure effower fi Jes,whofe two oppofite or contra? hatapa * ry fides are equall the one to the other. There are of paralldogrammcs fower ™Uelogrammf kyndes,afquareJafigureofonefidelonger,aRombusordiamond,andaRom- , . boides or diamond like figure,as before was fayde in the ^.definition of thefirif otparaMod* booke. Ofthefe fower fortes, the fquarc andthefigure of one fide longer are grammes* onely right angled Parallclogrammes; for that all their angles are right angles. And either of them is contayned (according to this definition ) vnder two right ly ties which concurre together ^nd caufe the right angle,and concaine the iame. Of which two lines the one is the length of the figure, & the other the breadth. The parallelogramme is imaginedto be made by thedraught or motion ofone efthe lines into the length ofthe other,As if two numbers ihoulde be multipli¬ ed the one into the other* AS the figure AB C D is a parallelograme, and is fayde to be contayned vnderthetwo right lines A B and A C,which contayne the right angle B A C,or vnder the two right lines AC and j ^ C D, for they likewife contayne the right angle A C D: of ~ ' 1 — — “ — which Julfnes the one,namely ,A B is the length , and theo- ther,namely,AG is the breadth* And ifwe imagine the line c ~ v AC tobe drawen or moueddir e&ly according to the legth Q»uii, q£ * Stmddt- fimtioHt r ? * The/econd^Boo^e of the line A B,or contrary wife the line A B to be moiled dire&ly according to the length of the line AC, you fhall produce the whole re&anglc parallelo- gramme AB CDwhichisfaydctobccontaynedofthem: eucn as one number multiplied by another produccth a plaine and rightc angled fuperficiall num¬ ber, as yc fee in the figure here fet, -where the number of fixe * % or fixe unities, is multiplied by the nupiber of fiuc or by flue vnities: ofwhich multiplication are produceijo^which number being fet downe and deferibed by his vn ides repre- fenteth a play ncanda right angled nurr^ter* Wherefore c- ucn as cquall numbers multiplcd by cqua^l numbers produce numbers cquall the one. to the other: fo reaangle parallelo- grames which are comprehended vnder equal Lines are equal the one to the other. 3o $ . . 1 ■ S.‘/tL ■ . . * ; ; - *- " ♦ ' . : * * 2. jl ’Jr, mu n-J ■ft In euery parattelogramme , one of thofi parallelogrammes , which foeueritbe, which are about the diameter , together* • 11 s- , • ft t V-V ‘ - lementesyi nomon. ■1 Thofe perticulcr parallelogramcs are faydeto be about the diameter of the parallelograme, which hauc the fame diameter which the whole parallelograme hath^And fupplementesareluchjwhich arc without the diameter ofthe* whole parallelograme. a s of the parallelograme ABCD the partial or perticulef paral - lelogrames GKCF and E B K H are parallelogramcs about the diameter, for that ech ofehem hath for his diameter a part o.f the diameter ofthe whole paraL* lelogramme. As C K and K B the pcrticuler diameters, are partes of the line C B,which is the diameter ofthe whole parallclogramme.And the two paralle- logrammes A E G K and KHF D,are fupplcmentes,becaufe they are wy thout the diameter ofth c whole parallelogramme,namely ,C B.Now any one ofthofe partiall parallelogram mes about the diameter together with the two fupple- mentes make a gnomon. As the parallelograme EB K H, with the two fupple- mentes A EG K andK HF Dmake the gnomon FGEfL Ltkewifetheparal* lelogramme G K C F with the fame two fupplemcntcs make the gnomon E H F G.Andthtsdiffinitionofagnomonextendethitfclfc, and is general! to all ky tides of parallelogrammes, whether they be fquarcs or figures of one fide lon¬ ger or Rhombus or Romboides. To be fliorte,ifyou take away from the whole parallelogramme one ofthe partiall parallelogrammes which are about the di¬ ameter Fol.6 f. ofEucUdes Elementes. amctet whether ye wilijthe reft of the figure isa gnomon. Campa eafterthelau propofitionof thefirftbookeaddeth this propofitio. Tvpo Squares bemggeuen , to adioyne to one of them a Gnomon equali to the other fquare .-which, for that as then it was not taught what a Gnomon is,1 there omitted, thinking that it might more aptly beplacedherc.The doing and demonftration whereat, is thus . proportion aided by (am* pane after the tajl proporti¬ on of the firfk hooks* Suppofe that there be two fquares A B and C D: vnto one of which, namely , vnto A £, it is required to addc a G nomon equali to the other fquare, namely, to C lD . Pro¬ duce the fide B F of the fquare AB di- reftly to the point £.and put the line F £ equali to the fide of the fquare CD. And draw a line from E to A. Now then forafmuch as E F Ais a rediangle trian- gle,therefore(by the 47. of the firft) the fquare of the line EA is equali to the fquares of the lines EF & FA. But the fquare of the line££ is equali to the fquare C2>,& the fquare of the fide FA is the fquare A ^.Wherefore the fquare of the line AE is equal! to the two fquares CD and A £.But the fides E F and F A arc (by the ai. of the firftj longer then the fide <sA £,andthe fide F A is equali to the fide £ £. W herfore the fides £ F and FB are longer the the fide A £♦ Wherefore the whole line BE is longer then the line A £,From the line £ £ cut of a line equali to the line A £,which let be B C.And (by the q-tf.propofition ) vpon the line B C deferibe a fquare, which let be BCGHtwhich lhalbe equal to the fquare of the line A £,but the fquare of the line A E is equal to the two fquares A B andD C.Wherefore the fquare B CGH is equal to the fame fquares. Wherfo re forafmuch as the fquare BCG His compofed of the fquare o 4 B and of the gnomon £ G A H , thefayde gnomon flialbe equali vnto the fquare C‘D:which was required to be done. An other more redy way after Pelitari us* Suppofe that there be two fquares,whofe fides let be iAB and£ C.It is required vnto the fquare of the line <*^££,to adde a gnomon equali to the fquare of the line's C.Setthc lines B and B Cin fuch fort that they make a right angle ABC, And draw a line fro to C.And vpo the line AB deferibe a fquare which let be A B ‘Z> £,And produce the line B A to the point £,and put the line BF equali to the line AC, And vpon the line B F deferibe a fquare which let be B F G H : which flialbe equal to the fquare of the line A C,whe as the lines B F and A Care equal land therefore it is equal to the fquares of the two hnes^Aand £C N°w forafmuch as the fquare BFG His made complete bv the fquare A £ D £ and by the gnomon £ £ GD,the gnomon F E CD flialbe * equal to the fquare of the line £ Cjwhich was required to be done. Conjlruttm, Vemottjlratio The feconcffiookg §&Thei fit heoreme . The iSPropofition* ff there be typo right lines , and if the one of them be deuided into partes hovoe many foeuer : the reBangle figure compre - hended vnder the two right lines js e quail to the reBangle y£- gures yphiche are comprehended vnder the line vndeuided% and vnder euery one of the partes of the other line . HH Vppofe that there be typo right lynes oyf and 3 C and let one of them y namely y 3 Che deui* dedatalladuenturesin the pointesDand E.Then I/ay that the reBangle figure comprehended Vn» jder the lines A3and 3 Cyis e quail Vnto the re Ban * if* figure comprehended Vnder the lines Jyand 3 '~D,oVnto the reBa?igle figure which is coprehen » ded Vnder the lines A and 3) Ey and alfo Vnto the _ ^reBangle figure which is comprehended Vnder the lines A and E C, For from the point e 3rayje Vp (by the u.of the firfi) vnto the right line 3C a perptndiculer line 3 EyO Vnto the line A (by the third of the fir It) put the line 3Ge* quail , and by the point G (by the $ 1 , of the firfi ) draw a parallel line Vnto the right line 3 C and let the fame be G Myand(by the felfe fame) by) points DyE^andCy draw Vnto the line 3 G the/e parallel lines ID E L and C H. TSloW then the parallelo- grame3 His equallto thefe parallelogrammes 3 KJD L}and E H.3ut the parallelograme 3 His equall Vnto that which is contayned Vnder thelinesAand3C. (For it is com* preheded Vnder the lines G3 O 3Cyand the line G 3 is equall Vnto the line A) And the parallelograme 3 Kjs equall to that which is contayned Vnder the lines A and 3 D: (for it is comprehended Vnder the line G 3 and 3 D}and 3Gise» quail Vnto A) And the parallelograme D L is equall to that winch is contayned Vnder the lines A and D E(for the line D that is/3 Gis equal Vnto A) And moreouer likewife the parallelograme E His equall to that which is contained Vnder the lines A o EC. VVberfore thd t whtch is compreheded Vnder ji lines A O 3C is equall to that which is comprehended Vnder the lines A O' 3 D-,0 Vn* toy which is compreheded Vnder the lines A and D Ey and moreouer Vnto that which is comprehended Vnder the lines A and E CJf therfore there be two right lines }and if \ the one of them be deuided into partes how many foeuer yt he reBan* Fol.dl. ofEuclides Elementes. gle figure comprehended Vnder the ttoo right lines js squall to the re&angle fi* gures ‘tohich are comprehended Vnder the line Vndeuided and Vnder euery one of the partes of the other linei'tobicb ‘tods required to be demonstrated* Becaufe that all the Proportions of this fecond booke for the moft part are true both in lines and in numbers, and may bedeclaredby both: therefore haue I haue added to euery Propofition conuenient numbers for themanifeftation of the fame. Andto the end the ftudiousand diligent reader4 may themore fully perceaue and vnderftand the agrementof this art ofc Geometry -with the fcience of Arithmetiquejandhow nere Sc deare fillers they are together,fo that the one cannot without great blemifh be without theother, 1 haue here alfo ioyneda little booke of Arithmetique written bv on c'Barlaam, a Greekc authour a man of greate knowledge* In whiche booke arc by the authour demonftrated many of the felfe fame proprieties andpafihons in number, which Euclideia this his fecondboke hath demonftrated in magnitude, namely .the firft ten pro- pofitidns as they follow in order. Which is vndoubtedly great pleafure to co- fider,alfo great increafe SC furniture ofkno wledge. Whole P ropofitiSs are fee orderly after the propofitiSs of Euclids, euery one of^/^iwcorrefpodent to the fame o£Euclide.An& doubtles it is wonderful to fee howthefe two cotrary kynds of quantity , quantity diferete or number, and quantity continual or magnitude (whicharethe fubie&es or matters of Arithmitique and Geometry ) fhoulde haue in them oneand the fame proprieties common to them both in very ma¬ ny pomts.and affc<ftions,although not in all. For a line may in fuch fort be dc- uided, that what proportion the whol e hath to the greater parte the fame {hall the greater part haue to the leCIc^ But that can not be in number. For a number cannot fo be deuidedjthat.the whole number to the greater part thereof, ftiall haue that proportion which the greater part hath to the leffe, as lordanevery playnely protieth in his booke of Arithmetike, which thynge Campane alfo fas we ftialiafterwardinthe9* booke after the 15* propofition fee) proueth* Andas touching thefe tenne firfte propofinonsof the feconde booke of Eu- clide,demonftratedby Barlaam in numbers,they are alfo dembftrated of Cam- pane after the i^propofition ofthe9- booke, whofe demonftrations I mynde by Godshelpe to fetforth when I fiiai come to the place. They are alfo demo- llrated of lordane that excellet learned authour in the firft booke of his Arith¬ metike. Inthc meane ty me I thougheit not amilfe here to fet forth the demon- ftrations of Barlaam, for that they geue great light to the feconde booke ofEu- clide,befides the ineftimable pleafureywhich they bring to the ftudious confidc- rer,Andnowto declare the firft Propofition by numbers. I haue put this exam¬ ple following. Take two numbers the one vndeuided as 74, the other deuided into what partes 37* deuided into 20. 10.?. and 2:which altogether make the whole 57. Then ifyou multiply the number vndeuided, namely, 74, into all the partes ofthe number deuided as into 20. 10. ?.and 2. you (hall produce 1480. 740* 3 70 . i^S.which added together make 273 8 :which felf numberis alfo produced if you multiplye the two numbers firft geuen the one into the other. As you fee in the exam¬ ple on the other fide fet. Barlaam, Barham, The fecondBoofy 74 Multiplication of the whole 1480 nuber vndeuided into the 740 partes of the whole num- 370 ber deuided. 148 •s 2738 - Multiplication of the one 74 whole number into the 0- 3 7 ther. y/e L ' ? -y 518 X * t " 7 "l— ” - 2 7 3 8 5 2 the number produced of the one-s whole number into the partes of the other whole number ^equall to ^the number produced of the fame whole into the other whole - So by the aide of this Propofition is gotten a compendious way of multiplication by breaking ofoneof the numbers into his partes: which oftentimes ferueth to great vie in working, chiefly in the ruleofproportions.Thedemonftration ofwhich propofition followeth in Barlaam.But firft are put of the author thefe principlesfoIlowin<*, <7 (principles. 1 . <zA number is fayd to multiply an other number: When the number multiplied is fo oftentymes added to it felfe, as there be vmties in the number, Which multiplied: Vo her by is produced a certame number Which the number multiplied meafureth by the unities Which are in the number Which mul¬ tiplied . 2 . And the number produced of that a multiplication is called apUine or fuperfciall number. 3 . *s4 fquare number is that which is produced of the mnltiplicatian of any number into it felfe. 4. Entry lejfe number compared to a greater is fayd to be a part of the greater, Whet her the lejfe mea~ fare thegr eater, or meafure it not. 5 . TfumberSiWhome one and the felfe fame number meafureth equally, that is, by one and the felfe fame num beY are e quail the one to the othet , 6. Numbers that are equemultiplices to one and the felfe fame number, that is, Which contayne one and the fame number equally and alike, are equall the one to the other. TbefirH Proportion, T tya numbers b eynggeucnjfth e one of them be deuided into any numbers how many foeuer: the playne orfuperficiall number Which is produced of the multiplication of the tWo numbers firft geiien the one into the other, (hall be equall to the ftpcrfciall tiu- bers Which are produced of the multiplication of the number not deuided into eusry part of the number deuided. Suppofe that there be two numbers A B and C. And deuide the number A B into certayne other numbers how many foeuer.as into A D,D £,and£ B , Then I fay that the fuperficiall number which is produced of the multiplication of the number Cinto the number zA B is equall to the fuperficiall numbers which are produ¬ ced of the multiplication of the number C into the nu- ber A 'Zhand of C into D £,and of C into £ B. For let F be the fuperficiall number produced of the multiplica¬ tion of the number C into the number A J3,and let GH be the fuperficiall number produced of the multipli¬ cation of Cinto Aid .-And let H I be produced of the multiplication of Cinto D £: and finally of the multi¬ plication of Cinto FB let there be produced the num- c A F G ber ofEuclides Ekmentes . Fol. 6^ ber/ ANowforafrniichas.^FmuItiplying the numberC produced the number F; therefore the number C meafureth the number/7 by the vnities which are in the nnm- ber A B. And by the fame reafon may be proued that the number C doth alfo meafure the number (7 //,by the vnities which are in the number?^ A and that* it doth mea¬ fure the number HI by the vnities which are in the nuber D F and finally that it mea¬ fureth the number IK by the vnities which are in the number E B .Wherefore the nu¬ ber C meafureth the whole number G K by the vnities which are in the number AB. But it before meafured the number F by the vnities which are in the number ABywher fore either of thefe numbers F and G’F'is equcmultiplexto the number C . But num¬ bers which areequemultiplices to one & thqfelfe fame numbers areequall the one to the other (by the tf.definitionJ.Wherfore the number Fis equadto the number G K. But the number Fis the fuperficiall number produced of the multiplication of the nu¬ ber C into the number A B : and the number Cj K is ccmpofed of the fuperficiall num¬ bers produced ofthe multiplication of the nuber Cnot deuided into euery one of the numbers A D,D E,andEB.l£ therefore there be two numbers geuen and the one of them be deuided &c. Which was required to be proued. The iJTbeoreme. The ?. . Eropofition , If a right line be deuided by chaunce > the nil angles figures which are comprehended vnder the whole and euery one of the partes , areequall to the fquare whiche is made of the whole . - V ppofe t hat the right line AB he hy chaunfe de- nided in the point C . Then 1 fay that the reHan * file figure comprehended Vnder Aft and B C to* gether "frith the reH angle comprehended "Vnder A B and AC is equal! Vnto the fquare made of A B,T)e* f cribs (by the 4.6,0 f the firH) Vpon A B a fquare A T) E B: and (by the $ 1 0} the fir ft) by the point Cdrafr a line paral * lei Vnto either of tbe/e lines A T> and B E,and let the fame be CF. Tfow is the ^emwJlratiS parallelogramme AE equallto the paralltlbgrammes A Fand C Ey by the firH of this books. But A E is the Jquaremadeof A B. And JF is the re cl angle parallelogramme comprehended "vnder the lines B A and A C: for it is compre¬ hended Vnder the lines V A and A C: but the line A V is equallvnto the line A B,Andlih"frife the parr allelogramme C Eis e quail to that "frhich iscontayncd "Vnder the lynes ABand B (for the line BE is equal Vnto the line AB.Wher - fore that "frhich is contayned "vnder B A and AC together "frith that "frhich it contayned "Vnder the lines AB and B C fis e quail to the fquare made of the line A B Jf therefore a right line be deuided by chaunce ythe re Ha file figures "frhich are comprehended Vnder the "frhole , and euery one of thepartes3are t quail to the fquare "frhich is made of the whole: ", frhich "fras required to be demonstrated. An other demonftration of Campane. ... R iii. Sup- A C B Id f TL Conjlruttion, ’ThefecondBooke Suppofe that tKe line AB be deuided into the lines AC} C Z>,and Di? .Then I fay that the fquare of the whole line A B, which let beAE BF,is equal to the redangle figures which are contayned vnder the whole andeuery one of the partes : fo r take the line ICwhich let be e- qual to the line Atf.Nowthen by thefirftpro- pofition the redangle figure contained vnder the lines A B and /<T,is equal! to the redangle figures contayned vnder the line K and althe partes of the line AB. But that which is con¬ tayned vnder the lines K and A B is equall to the fquare of the line A B , and the redangle figures contay ned vnder the line K and al the ^ G v 5 partes o£AB, are equall to the redangle fi¬ gures contayned vnder the line A B and all the partes ofthe line AB: for the lines B and K are equall: wherefore that is manifefl: whichwas required to be proued. Anexampleofthis Propofitionin numbers. Take a number, as 1 1 .and deuide it into two partesmamely, 7. and 4.’ and multiply x x .into 7 ,andthen into 4, and there lhalbe produced 77. and 44.‘both which numbers added together make r 2 1 .which is equall to the fquare number produced of the mul¬ tiplication of the number 1 1 .into himfelfe,as you fee in the example, •*', . . . . ^ * . \ ■ . ft4' . • t H - ? *% ^Multiplication of the whole- ’ intohispartes. A | Multiplication ofthewhole Mnto himfelfe. 11 77 44 1 21 11 1 1 11 1 1 121 the number produced ofthe-% whole into his partes* -equal to '-the number produced of the. whole into himfelfe. CqT- Barlaam. The demo nftration whereof folio weth in Barlaam, ThefecondTropofition. If a number geuen be decided into two other numbers : the fuperficiall numbers , Which arepro* disced ofthe multiplication ofthe Whole into either part, added together ,are equall to the fquare num¬ ber ofthe whole number geuen. Suppofe that the number geuen be A B : and let it be deuided into two other num¬ bers A C and CB. Then I fay that the two fuperficiall numbers , which are produced ofthe multiplication of A B into A C, and of A B into B C , thofe two fuperficiall num¬ bers (I fay) beyng added together, lhalbe equall to the fquare number produced of the multiplicand of the number AB into it felfe.For let the number <eA B multiplying it felfe produce the number Df Let the number A C alfo multiplying the number A B produce / of Euclides Elementes, Fol6\. B \i% 'F 4* produce the number £F:agayne let the numberCi? multiply¬ ing the felfe fame number AB produce the numberFG. Now1 foraftnuch as the number^ C multiplying the number zA B produced the number EF: therefore the number zAB meafu-* reth the number E F by the vnities which are in A C.Againe for- afmuchasthe number CB multiplied the number ^4 £,andpro duced the number F G = therfore the number A B meafureth the number FG by the vnities which are in the number Ci? .But the fame number AB before meafured the number E F by the vni- tieswhich are in the number AC. Wherefore the number A B pieafureth the whole number S G’by the vnities whcih are in A ^.Farther fo rafmuch as the number 2? multiplying it felfe pro duced the number/?: therefore the Humbert'S meafureth the --,C ^ number D by the vnities which are in himfelfe.Wherfore it mea % jp fureth either of thefe numbersmamelyjthe number £>,{and the number E G, by the vnities which are in himfelfe . Wherfore how multiplex the number D is to the number AB , fo multiplex is the number EG to the fame number AB. But numbers which are equemultiplices to one and the felfe fame number^re equal the one to the other. Wherefore the number *D is equallto the number E G .And the number D is the fquare number made of ^ the n urn ber A £,and the number E G is compofed of the two fu- perficiall numbers produced o£A B into BC, and o£ B Ainto A C. Wherefore the fquare numberproduced of the number <A B is equall to the fuperficial numbers,produced of the number A B into the number £ Cs and ofAB into AC,added together .If therefore a number be deuided into two other number s &c, which was required to be proued. 1 D i B be 3. The or erne, The 3. Tropofition . ff a right line be deuided by chaunceithe reBanglefigure com¬ prehended vnder the whole and one of the partes ,is equall to the reft angle figu> e comprehended vnder the partes , & "vnto the fquare which is made of the for ef aid part . Vppofe that the right linegeuen AB be deuided bycbaunce in the point C.Tben 1 fay tbat the re Handle figure compreheded Vnder tbe lines A & and B C is equall Vnto tbe re Bangle figure comprehended Vnder tbe lines A C and C B^and alfo Vnto tbe fquare which is made of tbe line B C. fDefcribe(by tbe 4.6, of the firfifvpon tbe line B Ca fquare CfD EB : and (by tbe fecond peticion)extendE Vnto F. And by tbe poitit A draw (by tbe 3 1 .of tbe firft)a line parallel Vnto either of thefe lines C B) and BE^and lettbe fame be AFJSLow tbe paraUdo* b c 4 grame A E is equall Vnto the parallelo* [~ 1 — grammes A B) and C E,And A E is tbe re Bangle figure comprehended Vnder |__ _ _ _ _ _ tbe lines A B and B C,For it is compre* E b p bended Vnder tbe lines A B and B E. %iif but Conflmtlm, Demnftmio Thefecond'Booke "fohich line BE if e quail Vnto the line S Ct jind the paradelograme A D is e* quail to that which is contayned Vnder the lines A C and C <B\ for the line 5) C is equall Vnto the line C B . And ID B is the fquare Tbhich is made of the lyne C B. VVherfore the re Handle figure comprehended Vnder the lynes A B and BC is equall to the re Bangle figure comprehended Vnder the lines A C and C 3 Ural* fo Vnto thefquare Tbhicb is made of the line 3 Ct If therfore a right line he de* uided by chaunce%the reBangle figure comprehended Vnder the tohole and one of the partes fis equall to the reBangle figure comprehended Vnder the parte f, andvnto the fquare ^hichis made of the forefayd part: H>hich 'to as required to he proued , An example ofthis Propofition in numbers, Suppofe a number,namely,i4.to be deuided into two partes 8,and 6. The whole number ^.multiplied into 8. one of his partes,produceth i i2:the partes 8. &<5. mul¬ tiplied the one into the other produce 48,whicn added to 6^( which is the fquare of 8. the former part of the number Jamounteth alfo to 1 1 1 : whiche is equall to the former fumme.Asyou fee in the example. r Multiplication of the whole" into one of his partes. r 14 8 O: j-the partes. ■ II2_ — "the number produced of the-. whole into one of hispartes- Multiplication of the one 8 part into the other. ' - 6 :>equa! to - 48 48 ■ J± Multiplication of the for¬ mer part into it felfe. 8 8 112 ^the number compofed of the one partinto the other, and- of the former part into him- felfe. *4. J BarUam* The demonftratton hereof folloveth in Barlaam, The third propofition. If a number geuen be deuided into tWo numbers: the fiperfciall number Which is produced of the multiplication of ih e Whole into one of the partes, ts equall to the/uperficiall number which it pro - duced of the partes the one into the olher,andtotbe fquare number produced of the forefayd part. Suppofe that the number geuen be v4 #,which let be deuided into two numbers A Cand C # .Then /fay that the fuperficiall number whiche is produced of the multipli¬ cation ofthe number^.# into the number;# C is equall to the fuperficiall number w'hich is produced of the multiplication of the number A C into the number C #,and to the fquare number produced of the number C '3. For let the number a A B muitipli- eng thq. number C B produce the number D.And let the number A C multiplieng the number CB produce the number E .Ftand finally let the number C B multiplieng him- felfe produce the number F G. Nowforafmuchas the number,//# multiplieng the ' - number ofEuclides Elements s. . 6 5* t 4r A number CB produced the huitibefi) .Therfore the number C B meafureth the number D by the vnities whiche are in the number A 5.Agayne forafmuch as the number^ Cmultipli- ed the numberCi?,and produced the number E F} therefore the number C B meafureth the nuber EF by the vnities which are m A C.Agayne forafmuch as the number C B multiplied it felfe and produced the number EG1-. therfore the numberC# meafureth the number F G by the vnities which are in it felfe. But as we haue before proued the felfc fame nuber CB mea¬ fureth alfo the number EF by the vnities which ate in the nu¬ ber A C,w herfore the number CB meafureth the whole num¬ ber EG by the vnities which are in the number AB, And it al¬ fo meafureth the number D by the vnities whiche are in the number A 2> .Wherfore the number CB equally meafureth ei¬ ther number,namely,the number ZJ^and the number EG. But thofe numbers whomc one and the felfe fame number mcafu- reth equally, are equall the one to the other. Wherfore the number D is equall to the number E G .But the number D is a fuperficiall number produced of the multiplication of the number AB into the number 2>C, and thenumberACis the fuperficial number produced of the multiplication of the nu¬ ber AC into the number CB, and of the fquare of the number CB. Wherfore the fuperficial number produced of the multi¬ plication of the number -^5 into the numbered is equal to the fuperficiall number produced of the number A Cinto the number CB, ana to the fquare of the number C .S.If therfore a number be deuidedinto two numbers.the fuperficiall nuber &c: which was required to be proued. S 1 E \ The ^Theorems. The 4.. Tropofition , If a right line be deuided by chaunce, the fquare whiche is made of the whole line is equal to thefqmres which are made of the partes, & vnto that rectangle figure which is compre* bended lender the partes twife. B yppo/e that the right lyne A © he hy chaunce deuided in the pointe C. ™| Then l Jay that the fquare made of the line A © is equall Vntoy [quarts which are made of the lines A CandC B, and Vnto the re Bangle figure contained Vnder the lines A C and C © tftife. Defer the (by y 46.0 f the first) Vpon the line AB a fquare ADE © land draft a line from B to fD,and(by the 3 1 .oft he h firft)by the point C draw a line parallel Vnto either of theje lines A Bland © E cutting the diameter © X) in the point G,and let the fame be C E.And(by the point G (by the felfe famefdraft a line parallel Vnto eyther of the fe lines A © and BE, and let the fame be H And forafmuch as the line CF is a, . parallel Vnto S.L CwJlrH&ion • the Vemonfiratio (eft theline AD,andvponthemfalletb arigbtlineft ft): therfore(by the 2 9, of the firft)the outward angle CG ft is equallVnto the inward and oppojite angle A ft ftJBut the angle Aft ft is (by the 5* of the fir ft) equali Vnto the angle A ft ft: for the fide ft A is equali Vnto the Jide A ft (by the definition of a fquare ). Wherfore the angle CG ft is equali Vnto theangleG ftC: wherfore( by the 6, of the fir ft )t he fide ft C is equali Vnto the fide C G.ftut C ft is equallVnto G and C G is equali Vnto Kfft: wberforeGJfjs equallVnto ft. VVherfore the figure CGKfift confifteth of four e equali ftdes.l fay alfo that it is a reHangle fi¬ gure. For for ajmacb as CG is a parallel Vnto ftK&vpon the falleth a right line Cft,therfore(byy 9. of the i,)tbe angles JfJB C^andG C ft are equal vnto two right angles /But the angle KfftC is a right angle givher fore y angle ftCGis alfo a right angle VV her f or e(by the 3 4 yfthe first )tbe angles oppojite Vnto them, namely yC G If, and G Kjft are right angles, Wherfore C G If ft is a re Han* gle figure. And it Was before proued that the fides are equali. VVherfore it is a fquare ^andit is defcribedvpon the line ft C And by the fame reafon alfo H Fis a fquare ,and is defcribed Vpon the line H Gjhat isy>* p on the line A C.VVberfore the f'quares Ff F and C If are made of t he lines A C and C ft And forafmucb as H the parade lograme AG is (by the 45, of the fir ft) e* quail Vnto the parallelogramme G E.And A G is that which is contayned V rider AC and Cftjor CGis equal Vnto C ft,wherforeG E is equali to that which is con» tained Vnder A C and C ft, V therefore A G audG E are equali Vnto that which is comprehended Vnder A CandC ft twife. And the /quay es H Eand C Jfaremade of the lines AC and C ft. VVherfore theft foure reHanglefigures HFflf^A Go and G E are equali Vnto the Jquareswbiche art made cf the lines AC and C ft,andto the reH angle figure which is comprehended Vnder the lines AC and C ft twife. ftut the reH * angle figures HF> C If, A G,and G Eare the whole reHangle figure At) Eft Which is the fquare made of the line A ftJWherf ore the fquare which is made of the line Aft is equal! to the fqmreswhicharemade of the lines A CandC fty and Vnto the reHangle figure which is comprehendedvnder the lines AC and C ft t wife. If therfore a r ight line be deuided by chaunce , the fquare which e is made of the whole line fis equali to the (quarts which are made of the partes^ Vnto the reHangle figure which is comprehended Vnder the partes twifeiwhich Was required to be proued * \{t- ration * :.u\ 5' v\ . > I fay that the fquare of the line AftisequaH Vrito the fquare s Wbiche art made of the lines A C and Cftjt? vnto the reitangle figure which is compnbe* ded Vnder the lines AC and Cft twife.Fpr the felfe fame difcription abiding for* 5 afmuch B 7 / / ofEuclidcs Elementes. FoL 6 6 . a/much as the line AB is equal l Vnto) Ime At>yy angle AB t> is(hy the of the firH) equal 7 Vnto the angle A T> B. And fora fmuch as the three angles of euery triangle areequalto two right angles (by the 5 2 of the fir ft). therefore) three angles of the triangle A BS)ynamely^ the angles A EDBfD B Afand B A Dyare e quail to two right angles.But the angle B AT> is a right angle therefore the angles remayning A B D>and A T) By are equallvnto one right angleiand they are equally one to the other ywher fore either of thefe angles A B Bdyzs' A Bfts the halfe of a right angle, And the angle BCG is a right angle , for it is equall Vnto the oppofite angle at the point A (by the 2 9. of the fir ft). V therefore the angle rtmaymng CG Bis the halfe of a right angle. Wherefore the angle CGB is equal Vnto theangle CBG: therefore alfo the Jide BC is equallvnto thefide CG.But BC is equall Vnto u G and C G is equal Vnto B Wherefore the fin gureC K^confiftetb of equall fidesiand in it is a right angle C Bl<f. Where » fore C Kjs a fquare, and is made of the line B C. And by the fame reafon H F is a fquare,and is equall Vnto that fquare which is made of the line A Q Where* fore C Kjnd El Fare fquares^and are equall to thofefquares which are made of the lines A Cand C B. Andforafmuck as AG is equall Vnto E G:and AG is that which is contaynedvnder A Cand C BJorGC is equal Vnto C Biwhere* fore E G alfo is equall to that which is coprehended Vnder A C and C B: where* fore AG and EG are equall Vnto that r eBangle figure which is comprehended Vnder A Cy and C B twife.AndCK^and H Fare equal Vnto the f quires which are made of A Q and (fB: wherefore C^HFjAG, and GE are equal Vn - to thofefquares which are made of A C,and CBjnd Vnto that re El angle figure Which is comprehended Vnder A Cand C B twife. But C Jf^ElFA GyandG E are the whole fquare A E which is made of A B , Wherefore the fquare which is made of A Bfts equall to the fquares which are made of AC and C By and Vn* to t he r eft angle figure which is comprehended Vnder A C and CB tWife: which was required to be demonstrated. Hereby it ismanifeft that the farallelogrames Vebich conftfl about the diameter of a fan art mtfi ^ Corollary* needes be /quarts. This proportion is of infinite vfe chiefcly infurde numbers.By helpeofitis made in theadditio Sc fubftra&ion,alfo multi plicatio in Binomials & refidu- als. And by hclpc hereofalfo is demonftratcd that kinde of equation, which is when there are three denominations in naturail order, or equally diftant, and two of the greater denominations are equall to the thirde being lefie Onthis propofition is grounded theextraftion of fquare roots.And many other things are alfo by it demonftrated* s.tj. An Balaam, fe The fecondBookg An example of this Tropofition in numbers . Suppofe a number namely, x 7. to be deuided into two partes 9. and 8. The whole number 1 7 .multiplied into him felfe,produceth 289. The fquare numbers of 9. and 8. are 8 1. and 64: the numbers produced of the multiplication of the partes the one in- to the other twife are 72, and 72: which two numbers added to the fquare numbers °t 9,and 8.namely,to 81. and 6 4. makealfo 3 8p.whichiscquall to the fquare number of the whole number 1 7 . As you fee in the example. The multiplication of the-', whole into himfelfe. r 17 17 {%: >-the partes of the whole n 9 17 28p r the number produced of the whole into himfelfe. The multiplication ofeche 9 part into himfelfe. 9 81 ^equall to 8 81 8 *4 <54 7a 7* - . • . .V. ‘ 289 "•the number compofed of cche The multiplication of the one part into the other 9 8 part into himfelfe.andof the one into the other twife. twife. 72 - - . . , .. J A * 9 8 - L 7» The demonftration wherof followcth inBarlaam, The fourth Tropofition . If a number geuen be deuided into tWo numbers : thejejuare number ofthewbole,is c/juatt to the fcjuare numbers of the partes, and to thefuperficiall number which is produced of the multiplication of the partes the one into the other tWtfe. Suppofe that the number geuen bet AH: which let be deuided into two numbers A C and CB. Then I fay that the fquare number of the whole number -/42?,is equall to the fquares of the partes,that is,to the fquares of the numbers C and CA,and to the fuperfkiali n umber produced of the multiplication of the numbers AC and CB the one into the other twife. Let the fquare number produced of the multiplication of the whole number AB into himfelfe be D . And let C A multiplied into himfelfe produce the number£-F; KndfB multiplyed into it felfe let it produce GH: andfi- nallyofthemultiplicatioofthe numbers A CandCAthe one into the other twife let there be produced either of thefe fuperfkiali numbers F G and Ft K . Now forafmuche as the number e^/Cmultiplyingit felfproduced the number .E A; therefore the num¬ ber zAC meafureth the number EF by the vnities which are in it felfe. Andforafmuch as the number CB multiplyed the number C A and produced the number F G : there- ofEuclides Elementes « Fc-Ldj. fore the number ^Cmeafureth the nuber F G by the vnities whiche are in the number CB. But it before alfo meafured the number SFby the vnities which arein it felfe. Where¬ fore the number A B multiplying the number -4Cprodu- ceth the number E G\And therefore the number EG is the fuperficiall number produced of the multiplication of the number B A into the number C, And by the fame rea- fonmay weproue that' the number G K is the fuperficiall number produced of the multiplication of the number A B into the number #C.Farther the number Z> is thefquare of the number A B. But if a number be deuided into two num¬ bers, the fquare of the whole number is equall to the two fuperficiall numbers which are produced of the multipli¬ cation of the whole into either the partes ("by the 2«Theo- reme.) Wherefore the fquare number D is equall to the fu¬ perficiall number E K. But the number EKis coinpofed of the fquares of the numbers A C and C B, and of the Superfi¬ cial number which is produced of the multiplication of the nuber A C and C B the one into the other twife:& the num¬ ber D is the fquare of the whole number AB. Wherfore the fquare nu mber produced of the multiplication of the num¬ ber A B into himfelfe, is equall to thefquare numbers of the partes, that is,to the fquare numbers of the nuber sAC and C2?,and to the fuperficiall number produced of the multiplication of the num¬ bers A C and C B , the one into the other twife. If therefore a number geuen be deui¬ ded into two numbers &c .Which was required to beproued. The 5. T beoreme . The jfPropofition, % If a right line he deuided into two equall partes^ & into two ynequall partes: the reUangle figures comprehended ynder the ynequaH partes of the whole ^together with the fquate of that which is betwene the feUiosJs equal to the fquare which h made of the halfe. mqyppof6 that the right line A B be derided i nto tWo equall partes in the Cydnd into two wequall partes in the point D.Tben I fay that the reSlangle figure comprehended Vnder AD and D B together With the fquare which is made ofC Dy is equall to thefquare which is made ofC B, De* ConftrutHon . Jcribe(by the 4 6. of the firfi) VpponCB a fquare y and let the Jame be C EFB, jfnd(hy the firfi peticion)draWe a line from E to B.And byy point Ddrawe (by the $ \tof the firfi ) a line pa • rallelvnto echo/ the fe lines CE and B E cutting the diameter B E in the point H} and ktey fame be ID C. And agayne(by the felfe Satj. 6-f ’ L 41 b \x -H ■ F b- A** 3* A V E, fame Dmenfiratio a TheJiconcNSookg fame) by the point H dr aWe aline parallel Vnto ecbe oftbefe lines A B and EFyand let the fame be KJ) \ and let Kj) be equall 'Vnto AB, Andagaine (by the ( elfe Jame )by the point A draw a line parallel Vnto either of thefe lines C L and B 0, and let the fame be AI\. Andforafmuch as (by the 43 ,of the firfl )tbe f upplement CH is equall to the fupplemet HF.put the figure ID 0 co* mon Vnto them both . VF'herefore the whole figure CO is equall to the whole fi * gure DF.But the figure CO is equallvnto the figure ALt for y line AC is equall Vnto the line C B. Wherefore the figure AL alfo is equal Vnto the figure ID F. But the figure C H common Vnto them both J/V her fore the whole figure AH. is equall Vnto the figures D L and D F. But A H is equall to that which is co* tayned Vnder the lines A D and D Bfor D His equall Vnto D B.And the fi¬ gures F D &DL are the Gno - tnon MKXynerfore y Gno* A mon MNX is equall to that Which is contayned Vnder A D and V BjPut the figure LG co • mon Vnto them both^which is e* qual to the fquare Which is made of C (D, VFherefore the Gnomo M 2^1 X and the figure L G art equall to the rectangle figure co - prehended Vnder X D and D B and Vn(o the j quart which is made ofODjBut t he Gnomon M TL Xfand the figure L G are the whole fquare C E FB%whick is made ofBC.VF here fort the reft angle figure comprehended Vnder A ID and 1) B, together with the fquare which is made of CD yis equall to the fquare Which is made ofCB, If therefore a right line be deuided into two equall parts > and into two Vnequall partes yhe re ftangle figure comprehended Vnder the Vn» equall partes of the whole ^together with the fquare of that which is betWene the /eft tons, is equall to the fquare which is made of the halfe: which Was requia redtobeproued , This Propofitionalfoisofgrcate vfein Algebra. By it is demonftrated that equation wherein the grcatell and IcaitWe&es or numbers are equall tj the middle. r M K L M / H / AS P An example of this propofition in numbers. Take any number as 20: and deuide itinto two equall partes xo-and 10. and then into two vnequall partes as 15. and 7. And take the differcce of the halfe to one of the vnequall partes which is 5. And multiplythevnequallpartcs,thatis,i? andy.theone into the other,which make pi.take alfo the fquare of 3 . which is p. and adde itto the forefaydenumberpuandfolhallthercbe made xoo. Then multiply the halfe of the whole number into himfelf, that is, take the fquare of 10. which is 100. which is equal to the number before produced of the multiplication of the vncqual parts the one in¬ to the othcr,& of the difference into it felfe which is alfo 1 oo,As you fe in the example* The of Euctides Elmtntes* The whole euefl ^ ao fio iuy» Multiplication of the vn¬ equall partes the one into the other* Multiplication of the dif¬ ference into it felfe. Multiplication of the half into it felfe* io xo 1 ^ ^-the vnequall partes w 3 f^The difference of ohe of the vnequal partes to the halfe. X *1 7 9* Pl 9 j iOo 3 io io too ' the huber coiiipofed of the mul¬ tiplication of the vnequal partes the one into the other* & of the difference into it felfe ■cquall to H X the number produced of the halfe into it felfe. The demonftration 'wher°ffDllowech in Barlaam, The fifth proportion. If an earn number be derided into Wo e quail partes^and againealfo into Wo vnequall partes: the fuperfit'iull number Which is produced of the multiplication of the vnequall partet the one into1 the other together with the fquare of the numberfet betwene the parts , is equal to the fquare efhalft the number. ■Suppofc that^Abeaneuen number : which let be deuidfcd into two equal! numbers A Cand CBy and into two vnequall numbers A Jf &nd D B. Then 7 fay, that the fquare number which is produced of the multiplication Ofthe halfe number CJ3 into it felfe Js equal! to thefu- perficiall number produced of the multiplication of the vnequall numbers A I) and D B the one into the other, and to the fquare number produced of the number C D which is fet betwene the fay de vnequall partes. Let the fquare number produced of the multiplication of thb halfe number CB into it felfe be E . And let the fuperfU ciall number produced of the multiplication of the vne-, qual nuber s A D and D B the one into the other,be the number FCj-. and let the fquare of the number DC which is fet betwene the partes be G H, Now forafmueh as the number B Cisdeuided into the numbers BD and DCj therforethe fquare of the number 2? C,that is, the num¬ ber E,is eq uall to the fquares of the num bers B D and t) C3andto the fuperficiall number which is compofed of the multiplication of the numbers B D and t> C the one into the other twife,f by the 4.propofition of this boke) Let the fquare of the number E D be the number K L:& let N X be the fquare of the number D C: and finally of S.iiii. the 3 b -.'D 4 \ 6" b * C • 4 ; i ■ 4 Ou A* M.H %% 4 -f 4 i. n m E the multiplication of the numbers BD and D C the one into the other t\vife,let be pro duced either of thefe number's L M and M 2^. Wherefore the whole number K X i$ equall to the number £ Andforafmuch as the number BD multipliyngit felfe produ¬ ced the number K L, therefor it meafureth it by the vnities which are in it feife« More ouer forafmuch as the number CD multiplying the number B D produced the num¬ ber L A -/, therefore alfo D B meafureth L M by the vnities which are in the number C Z>: but it before meafured the number K L by the vnities which are in it felfe, Where- fore the number D B meafureth the whole number KM by the vnities which are inC B . But the number CB is equal! to the number C A. Wherefore the number 'DB mea¬ fureth the number K A4 by the vnities which are in C A. Agayne forafmuch as the nu- ber CT) multiplivng the number D B produced the number A/ 2W therefore the num¬ ber D B meafureth the number M 7^ by the vnities which are in the number CD -but it b efore meafured the number KM by the vnities which are in the number ^C.Wher fore the number B D meafureth the whole number KN by the vnities which are in the numbers D. Wherefore the number FG is equall to the number if iV.For numbers which are cquemultiplices to one and the felfe fame number,are equall the one to the other.ButthenumberG’i/ is equall to thenumbcrNX: foreither ofthem is fuppo- fed to be the fquare of the number C D.’Wherefore the whole number KX is equall to the whole number F H, But the number K X is equall to the number E. Wherefore alfo the number F His equall to the number £.And the number F His the fuperficial num¬ ber produced of the multiplication of the numbers and DB the one into the o- ther together with the fquare of the number DC, And thenumber £ is the fquare of the nutnberC£.Wherfore the fuperficiall number produced of the multiplication of the vnequal partes AD and DB the one into the other, togetherwith the fquare of the nu- ber D C which is fet betwene thofe vnequall partes, is equall to the fquareof the num¬ ber C 2?, which is the halfe of the whole number AB. If therfore an euen number be de- uidedinto two equall partesAx* which was required to beproued. 1 I The 6fTheoreme, The 6,cPropofition. ' - • . 3 - ' If a right line be deuided into typo equal partes, and ifvnto it be added an other right line direBly,the reB angle figure con* tayned vnder the whole line with that which is added,&the line which is addedt together With the fquare which is made of the halfe , is equall to the fquare which is made of the halfe line and of that which is added as of one li ne. '■ ' ! • • in • f-j' is ■< C1. Hi Vppcfe that the rigbte line A’Bbe deuided in - to two equall partes in point C : O' let there added Vnto it an other right line J)rBdirecHy}that is tojayynhich being ioyned Vnto A make both one right line A Then I fay, that the rectangle figure compre bended )mder A® and ' • .. i..!i .'mSfcsih. ofEuclides Elementes. FoL 6p. gether "frith the fquare whiche is made of BC is equall to the fquare wU'chefit made ofD C.DeJcribe( by the 46 .of the i ,)vpon CD a fquare C E FD,and (by the firfi petition )draW aline from D to E:and (by the tuoftbe first) byy point B draw a line parallel Vnto either of thefe lines EC&*DFy cutting the diame * ter D Em the point H^and let the fame be BG,&(by the felffitme )by y point EL draw to either of thefe Itnes A T) andEFa parallel line M: and moreouef by the point e A drawe a line pa a rallelto either of thefe lines CL and D M: and let the fame be A JfAnd forafmuch as AC is e quail Vnto C B^therfore (by the 5 6, of the firfi) the figure A Lis equal Vnto the figure C EL But(by the 43 .of the firfi )C His equal Vn* the figure H F, "frherfore A L is equall Vnto FL F. Tut the figure C M common to them both^wber fore the whole line A Mis equal Vnto the gnomon N X OjBut A Mis that which is contaynedvnder AT) andD B: for D M is equal Vnto TfBi ‘ frherfore the gnomon TsL % OJs equall Vnto the reBangle figure contained Vn* der A D and D BJPut the figure L O common to them both/frhich is equall to the fquare "frhich is made ofC B , Wherefore the reBangle figure which is con* taynedvnder AD and D B together with the fquare which is made ofC Bis e* quail to the gnomon IflCO^and Vnto L Gt But the gnomon NXO and L G are the whole fquare CE ED which is made ofC D4 V fiber fore the reBangle figure contaynedvnder A D and D B together with the fquare which is made ofC B is equall to the fquare which is made ofC D.Iftherforea right line be de* aided into two equall partes , and Vnto it be added an other right line direBlyi the reBangle figure contayned Vnder the whole line with that which is added9 and the line which is added ^together with the fquare which is made of the halfe , is equall to the fquare which is made of the halfe line and of that which is added as of one line : which Was required to be demonstrated. By this Propofition(be Tides many other vfcs) is in Algebra dcmonftrated that equation wherin the two leffe numbers be equall to the number of the grea^ tell denomination* An example of this propofition in number s4 Take any euen number as 1 8 .and adde vnto it any other number as 3 .which mafe in all 2 1 .And multiply 2 1 , into the number added, namely, into 3 ♦ which maketh 6 3 . Take alfo the halfe of the wholeeuennumber,thatis,of i8.whichis p. And multiply p. into it felf which maketh 8 1 .which adde vnto 63 .(the number produced of the whole cuen number jand the number added into the number added)and you fhal make 144., AC 3 i> ThefecondTlooke Then a dde p.thehalfe of the whole euen number vnto 3 .the number added which ma- keth 12 .And multiply i2.intoitfeIfe,thatis,take the fquare of 12, which is 144.WWCI3 is equall to the number compofed of the multiplication of the whole number and the number added into the nu m ber added,and of the fquare of the number added , which is alfo 144. As you fee in the example. ’•The whole euen number.-^ The number added. The halfe of the whole. The number added. 2 1 the number compofed of the whole number^ the number added. 9 3 Multiplication of the whole & the number ad- dedintothe number ad¬ ded. 12 the number compofed of the halfe andof thf number added. 2 1 3 63 *3 81 Multiplicand of thehalfe into it felfe. Multiplicand of the halfe and the number addedin to it felfe. 9 9 x44 81 it 12 24 12 Lx44 ■'■the nuber compofed ofthe whole- and the number added into the number added and of the fquare of the halfe ■equall to '"the fquare number made ofthe-’ number compofed of the halfe and the number added. The demonftration wherof folio weth in Barlaam* The fix t Tropofition. If an euen number be deiiided into two equall numbers, and -vnto it be added jbme other nttmheri the fuperfcia/l number Which is made ofthe multiplication ofthe number compofed ofthe whole num¬ ber and the number added, into the number added 5 together With the fquare ofthe halfe number, is equall to the fquare of the number compofed ofthe halfe and the number added. Suppofethat^2?bean euennumber.andletitbedeuidedinto two equall num¬ bers AC and CB : and vnto it let there be added an other number B D. Then /fay that the fuperficiall number produced ofthe multiplication ofthe number A'T> into the number D/? is equall to the fquare ofthe number CD, For let the fquare number of the number C D be the number £,and let the fuperficial number produced of the mul¬ tiplication of the number A D into the number D B be the number F C.-and finally let the fquare number of CB be the number G H. And forafmuch as the fquare of the nu- ber CD is (by the 4.propofition Jequall to the fquares of the numbers D B and BC to¬ gether with the fuperficiall number which is produced ofthe multiplication ofthe numbers DB and BC the one into the other twife. Let the fquare of the number BD be the number KL: and let the fuperficiall numbers produced ofthe multiplication ofthe numbers D B and 2?C the one into the other twifebe either of thefe numbers LM ofEuclides Elementes. FoL jot LM and M N; and finally let the fquare of the number 2? Cbe the number NX. Wher- fore the whole number K-Afhall be equall to the fquare of the number C D. But the fquare of the number C D is the number E. Wherfore the number K X is equall to the number F. And forafinuch as the number B D multiplieng it felfe produced the num¬ ber K L: therfore the number B D meafureth the num¬ ber K L7 by the vnities which are in it felfe, but it alfo meafureth the number L M by the vnities which are in the number CB, Wherfore the number D B meafureth the whole number K M by the vnities which are in the number C D. The nuber D B alfo meafureth the num¬ ber M N by the vnities which are in the number C B: & the number CB is equall to the number C A by fuppo- fition. Wherfore the number-D 2? meafureth the whole number K N by the vnities which are in the number A J>.Butthenumber2)2? doth alfo meafure the number F (7 by the vnities which are in the number AD : for by fuppofition the number F Cj is the fuperficiall num¬ ber produced of the multiplication of the numbers A D and -OF the one into the other* Wherfore the num¬ ber FCis equall to the number KN. But the number HG is equall to the number N X : for either of them is thefquare number of the number CB, Wherefore the whole number i7 His equall to the number KX: and the number KX is proued to be equall to the number F.W herfore the number F f/fhall alfo be equal! to the number E, And the number F A? is the fuperficiall num her produced of the multiplication of the numbers^ %) and DB the one into the other,together wyth the fquare of the number CBi and the number Fis the fquare of the number CD , Wher¬ fore the fuperficiall number produced of the multiplication of the numbers a/f D and J> B the one into the other, together with the fquare ofthe number CB ^ is equal! to the fquare of the number C D. 2f therfore an euen number &c* The jJTheoreme , The yfPropofition , ffa right lyne be deuided by chaimce, the fquare whicbe is . made of the whole together with the fquare which is made of one ofthe partes Js equall to the rectangle figure which is co<* tayned vncler the whole and the [aid parte twifie 3 and to the fquare which is made ofthe other part Fppof'e that the right line A B he deuided by chaunce in the point C, Then I fay that thefquare Which is made of A together With the fquare which is made ofB Cfs equal Vnto the r eh angle' figure which is contained Vnder the lines ABandBC twifeyand Vnio the fquare ‘ Which is made ofj. C/Dejcribe( by the 46 .ofthe firfi) Vppon A B a fquare A &)E B>arid make complete the figure .And forafinucb as (by the 45, ofthe fir [l ) ihe figure A G is equall vnto the figure G EfFut the figure C F common to the T, ' both] v 3 c b* *1. r r > * - ♦ _ Xjf c . 9 1 49 _ r 1 O « H i/A 2 r I E F K *1 he fecond'B coke both: Ibher fore the 'tobole figure A Fis equall to the Tbhole figure C B. VVher* fore the figures AFatid C E are double to the figure A F.But the figures jfF and CE are the gnomon Ffh M,and the fquare CF: "tober fore the gnomon L M,and the fquare C F is double to the figure AF. But the double to A Fis that Tbhicb is con* tayned Vnder ABandBC ffrife, for BE is e* quail Vnto B C W herfore the gnomon FfL M and the f quart C F is equal! Vnto the rebiangle figure contayned Vnder A B and B C twifefPut the figure DG commonVnto them both , "tohich is thefquare made of yi Ct Wberfore thegno* mon l^L M and the fquares B G and G ID are equal Vnto the rebiangle figure which is contain ned Vnder AB & BCtwife9(sr Vntothe fquare which is made of A C.But the gnomon the fquares BG%<sr DG are y “tobole fquare BAD Ey(sry part or fquare C F9which fquares are made of the lines AB & of BCyherf ore y fquares which are made of A B O' B Care equal i Vnto the re biangle figure which is contayned Vnder ABandB Ctwife9andalfa Vnto the fquare of A C.Iftherfore a right line be deuided by chaunceithe fquare whichis made of the whole together with thefquare which is made of one of the partes, is equall to the reblangle figure which is contayned Vnder the whole and the f ay d part twi[e9and to the fquare Which is made of the other parte : whiche Was required to be demonstrated . Fluflates addeth vnto this Proposition this Corollary* The [quarts of Wo vn equall lines do exc cede the reft angle figures contayned vnder the fitid Unit by the fquare of the exceffe voherby the greater lyne excedeth thclejfe. For if the line oA B be the greatcr,and the linci? C the lefle.it is manifeft that the fquares of AB andi? C are equall to the redangle figure contayned vnder the lyncs A B and rB C twife, and moreouer to the fquare of the line A Cwherby the line A B cxcc- derh the line BC, By this propofition moft \ronderfully vas found out the extra&ion of roote fquares in irrationall numbers,befide many other ftraungethinges. An example of this propofition in numbers . Take any number as 1 3 .anddeuideitinto two partes asint04.Sc p. Take thefquare! of 13. which is i6p, take alfo the fquare of 4. which is itf.andadde thefe two fquare* together which make 185 .Then multiply the whole number 1 3 . into 4. theforefayde part twife,and you fhall produce 5 2. and 5 2 : take alfo the fquare of the other part, that i s,of p . wh ich is 8 r .And adde it to the produdes of r 3 .into 4.twife,that is, vnto 5 2 .and 52. and thofe three numbers added together fhall make 185. whicheis equall to the number compofed of the fquares ofthc whole and of one of the partes, which is alfo 1 85 .As. you fee in the example* ofEuclides Elementes. Fol.yu 'The whole >3 it) »-partes of the whole* Multiplication of the whole into it felfe. n *3 39 13 1 6p 1 6$ 1 6 Multiplication of one of the partes into it felfc. *< ** • 4 4 id n 4, 185 'The number compo fed of the fquares ofthe whole, and of one of the partes Multiplicationof the whole into the forefayde part twife. 52 53 4 5 * Si nequall to 5* Multiplication of the o- ther part into it felfe* 9 9 8i 585 _ the number compofedof the whole into the fore- faid part twife,and ofthe fquare ofthe other part. The demonftration ^heroffolloweth In Barlaam. The feuenth proportion. tfa number be deuided into two numbers: the fquare of the "Whole number together With the fquare of one of the partes, is equallto the fuperficiall number produced of the multiplication of the Whole number into theforefaid part twife, together With the [quart of the other parti Suppofe that the number tAB be deuided into the number^ a^/Cand CBt Then 1 fay that the fquare numbers of the numbers BA and AC are equall to the fuperficiall number produced of the multiplicatio of the number B A into the number AC twife, together with the fquare of the number B C.For forafmuch asfby the 4.of this booke) 3 the fquare of the number AB is equall to the fquaresofthe numbers B Cand CA,znd to the fuperficiall number produced of the multiplication of the numbers *3 C and C A the one into the other twife: adde the fquare of the number C common to them both. Wherfore the fquare ofthen umbered .5 together with thefquare of the num¬ ber AC is equall to two fquares of the number A C and to one fquare of the number C By and alfo to the fuperficiall number produced of the multiplication of the numbers B C and C A the one into the other twife. And forafmuch as the fuperficial number pro ducedofthe multiplication of the numbers B A and C A the one into the other once, 5 is equall to thefuperficiall nuber produced of the multiplication of B C into C A once, and to the fquare of the number C A(by the third of this booke Jbtherforc the number produced ofthe multiplication of B A into *AC twife is equall to the number produ¬ ced of the multiplication ofB Cinro C^twife,andalfo to two fquaresofthe number C A.Addc the fquare number ofSCcommon to them both.Wherfore two fquares ofthe number C and one fquare of the number C2? together with the fuperficiall number T.iij. pro- A ■«» tP duced of the multiplication of B Cinto C./? twife are equall to the fuperficiall number produced of the multiplication of the number BA into the number AC twife toee- ther with the fquare of the number C B -Wherfore the fquare of the number A B toge¬ ther with the fquare of the nubcr AC is equal to the: fuperficial nuber produced of the multiplication ofthe number 5 -4 into the number A C twife, together with the fquare of the number CB. If therfore a number be deuidedinto two numbers &c. which was required to be demonftrated. b f 6 4 the fquare ofthe whole % 2 5 fthe fquare of the part j1Isut“4 8 y ^4 the fquare ofthe whole AJS 25 the fquare ofthe part A C 80 the fuperficiall number 9 the fquare ofthe other part B C §7 l 4° 4° 80 the fiiperficial number produced of the multiplication of the whole into the part twife $ . 9. r the fquare of the other par t. 7* he Theorems. The ftSPropoftidn, If a right line he deuided by chance foe rectangle figure com** prehended Under the whole and one ofthe partes foure times, together With the fquare which is made ofthe other parte, is equall to the fquare which is made ofthe whole and the for e^ Jaid part as of one line. - ■ Vppofe that there he a cert ay ne right line A and Jet it he deuided hy chaunce in the point C. Then 1 fay that the re hi angle figure comprehended Milder A B and BC foure tymes together M?ith the fquare Mohichismade of Aids equall to the fquare made of A B and BCas of one lin/, Extend the line A B (hythefecondpeticion). And (j (hy the third ofthe firft) Mnto C B put an equall lyne B And (hy the 46.of the firft) deferihe Mppon AD a fquare AEF D.And deferihe a double figure . And forafmuch as CB is equall VntoBDftttt CB is eqmlhnto GJ^ (by the^of the firft) and like wifeBD ofEuclides Elementes . Fol.Ji; N, ,/Y1 M / G /A K \X 0 / / V P ' 1 _ i ts equallvnto therefore G Ifalfo is equally nto h(K:and by the fame reafonalfo B is equall Vnto %0. And forafmuch as BC is equallvnto B and G Kgvnto LfKy tberfore (by the ^ 6. of the fir H) the figure C is equall Vnto the figure JfiJD}and the figure G %ts equall Vnto the A c "B i> figure (RJSl, (But (by y 4i,ofy ' u)the figure C Kjs equall Vn to the figure BJSL: for they are the fupplementes of the parallelograme C 0. VVher* fore the figure D alfo is e* quail Vnto the figure ISL Bj VVherefore the/e figures 2) KJCKJG BJBJSfyare equall the one to the other .Where* forethofe foure are quadru * pie to the figure C^. Agayne forafmuch as C B is equal Vn* ^ M L F to BDjout BID is equaUvn* toBKjcbatisjmtoCG.And C Bis equall vnto G K^that is Vnto G B: tberfore C G is equal! Vnto G B. And forafmuch asCG is equalhnto G B^and B BJs equall Vnto BJlytherefore the figure A G is equallvnto the figure MB ^and the figure B Lis equall Vnto the figure Bfd.But the figure MB is(by the 4% ofthefirU) equalhnto the figure F Lfor they are the fupplementes of the paratlelogramme M L: wherfore the figure alfo A G is equall vnto the figure BJ^. Wherfore the fe foure figures A GyMBfB L,and BJd are equall the one to the other: wherfore thofe foure are quadruple to the figure A G.And it is proued,that thefe foure figures CJfififfi) G B,-)fA.'HAre quadruple to the figure CKJ/ A her fore the eight figures whid > contayne the gnomon S T V^are quadruple to the figure A l\. And forafmuch as the figure A IQ is that which is comay ned Vnder the lines A B and B D,for the line B KJs equallvnto the line B D : tberfore that whiche is contayned Vn * derthe lines A B and B V foure tymes is quadruple Vnto the figure A And it is proued that the gnomon S TV is quadruple to AAf Wherfore that which is contayned Vnder the lines A B and BD foure tymes is equallvnto the gnomo S T VJBut the figure X Hwhich is equall to the fquare made of AC common Vnto them both.Wherfore the re dangle figure comprehended Vnder the lines A B and B D foure tymes together With the fquare which is made of the line A Cfis equall to the gnomon ST Vyand Vnto the figure X H. But the gnomon $ 'TV: and the figure X Hare the whole fquare ABF D, which is made of A T> : wherfore that which is contayned Vnder the lines A B and B D foure times together with the fquare which is made of A C, is equall to the fquare which is Tmu, made made of A 7). But B D is equalhmtoB C.Wherfore thereBangle figure con* tajnedfoure tymes Vnder A ’Band B C together tyith the {quarts which is made °f A Cjs equall Vnto the fquare Trhich is made of A ID, that isyVnto that ftbiche is made of A B andB C as of one line . If therefore a right lyne be deuided by chaunce , the relf angle figure comprehended lender the Tbhole and one of the partes foure tymes ^together faith the fquare which is made on the other part,k equal! to the fquare Tabid is made ofthelphole and the forefaid party as of one line>D?bicbl!>as required to be demonstrated- ; i An example of this Propofition in numbers. Take any number 3 s iy.anddeuideit into two partes,as into 6.and n. And multi¬ ply i7.into5.namelyoneofthepartesfouretymes,andyoufhallproduce 102. 102* 102. and io 2.Takealfo the fquare of 1 1. the other part,which is 121: andaddeitynto the foure numbers produced of the whole 17. into the part 5.foure tymes,&you ihall make <> 29. Then adde the whole number iy.to the forefaid part 6. which make 23 . Ss, take the fquare of 23* which is 5 29. which is equall to the number compofed of the whole into the fayd partfoure tymes, aud of the fquare of the other part, which num¬ ber compofed is alfo 5 zp. As you fee in the example. f - The whole. 6 102 j j partes of the whole Multiplication of the Whole into one of the partes foure times. X c.;:-L ~ '' - ■: Multiplication of the o- ther par t into it felfe. Addition of the whole in¬ to the part. Multiplication of the nu- ber copofed of the whole and the forefaid part into it felfe. 17 6 102 17 6 102 17 6 102 1 r I r I I 1 1 !I2I 17 6 23 2 i 23 \6 9 $29 ioz 102 102 102 121 5 29 • | 1 : fv' f . ' • ; ■ f ;v V h • c ' | the number compofed of the whole into one of the partes foure tymes, & of the fquare of the other part v equal to - ■> s-' the fquare of the number co- ’ pofed of the whole & the fore faidpart. * The ofEuclides Elements* . The demonftrationvheroffoilovethin Barlaam. The eight proportion. Fol. 73. D If 4 number be derided into typo numbers, the fuperficiall number, produced of the multiplication of t he whole into one of the partes foure tymes, together With the fqtiare of the other parte, is equall to the fquare of the number tempo fed of the whole number and the forefay d part. Suppofe that the number AB be deuided into two numbers AC and C B. Then / fay that the fuperficiall number produced of the multiplication of the number A B in to the number C B foure tymes together with the fquare of the number A Cfs cquall to the fquare of the number compofed of the numbers AB & CB. For vnto the num¬ ber 2? Clet the number 2? D be equall. Now forafmuch as the fquare of the number AD is equal to the fquares of the numbers A B and 2> D,Sc to the fuperficiall number produced of the multiplication of the numbers AB Sc 2 B D the one into the other twifef by the q-.of this booke) And the numb er 2?Z>is equall to the number B C\ therefore the : fquare of the number AD 2 is equall to the fquares of the numbers A B and B C, and to the fuperficiall number produced of the multiplication of the numbers AB and SC the one into the other twife.But the fquares of the numbers A B and B C are e- quall vnto the fuperficiall number produced of the multiplication of the 6 numbers A B and B C the one into the other twife, and to the fquare of A C(by the former propofition) Wherfore the fquare ofthe number ADis cquall to the fuperficial number produced of the multiplication of the nu- bers A S and B C the one into the other foure tymes, and to the fquare of [ the number A C.But the fquare of the number A D is the fquare ofthe number compofed ofthe numbers .^2? and 2? C: for the number# D is e- qual to the number B C. Wherfore the fquare of the number compofed of thenumbefs AB andS C is equall to the fuperficiall number produced ofthe multiplication of the numbers A B and B Cthe one into the other foure tymes, & to the fquare of the num¬ ber A C./f therfore a number be dcuidedinto two numbers, &c- 8 2 1 6 v___ 8 2 1 6 8 2 1 6 ~y — 8 2 1 6 6 6 64. the fuperficiall number produced of the multipli- _N cation ofthe numbers AB and B Cthe oneinto the other foure tymes. 4 3 6 the fquare of A C, 10 10 too 2 1 100 > thejquare of the number compofed o iAB and B C. the fuperficial nuber produced ofthe multiplicatiS made 4*times the fquare number of A C, ConHmlHon. DemonHru-* non. *Ihe fecondUfwhp Thep.Theoreme. ThepfPropofitm. ' ' ' - 1 1 ' ' ■ • '-'ey >\ I j'< Sfe? If i right line be dcuided into two e quail partes, and into Wo lone quail partes, the fquares which are made of the*vne~ quail partes of the whole, are double to the fquares, which are made of the halfe lyne^and ofthatlyne which is betwene the feUions . Vppofe that a certayne right line A 3 he dtuided into two equaS partes in the pointe C, and into t"Wo Vnequall partes m the pointe <D. Then I/ay t bat the Jquares "which ar e made of the lines A F) and S) 3 .are double to the/quarts whiche are made of the lynes A C and C JD.For(by tfe 1 \, of the fir ft )er eel from y point C to the right UneABa perpendiculer line C E. And let C E(by the $,of the fir ft) be put e quail Vnto either ofthefe lines ACzsrCB: and(by the fir ft peticio) draw lines from A to E, and from E toB.And (by the 5 1 . of the fir ft) by the point l D dra"W Vnto the line EC a parallel lyne,and let the fame be IDF: and (by the felfe fame) by the point E dra"W Vnto A3 a line paralleled let the fame be FG , And ( by the first petkion)draw a line from A to F, And forafmuch as AC is e* quail Vnto C Ejberforefby the 5 of the fir ft) the angle EACis equal Vnto the angle C E A And forafmuch as the angle at the point C is a right angle: therfore t/c angles remay n’wgE A f and A E C3aree quail vnto one right angle, "Where* fore eche ofthefe angles EAC and AECis the halfe of a righ t angle. And by the fame reafon alfoecbe ofthefe angles EfB Cand C E Bis the halfe of a right angle. VV her fore the "whole angle ABB is a right angle, And forafmuch as the angle G E F is the halfe of a right angle 3but EG F is a right angle. For (by the 2 9 of the fir ft )it is equall Vnto the in"Ward and oppofite angle, that is/Vnto EC 3: "wherfpre the angle remay ning E F Gis the halfe of a right angle ,VVhere * fore( by the 6 .1 common fentence )tht angle G EFis equal l Vnto the angle EFG, yVherforeal/o (by the 6.0ft he f it j fifth efid e EG is equdllvnto the fide F G,A# gains forafmuch as the angle at the point Bis the halfe of a right angle fut the angle F D B is a right angle for it alfo(by the 29 .of the firfi ) is e quail V nto the irtWard ofEuclides Element# s . inwarde and oppofte angle E C E. Wherefore the angle remayning EFT) is the halfe of a right angle . Wl:erfore the angle at the point E is equall lento the angle E EE. Wherfore ( hy the #>. ofthefrH) the fide E) F is eqnall lento the fide E) E. Andforafmuch as AC is equal l lento C E> therfore the fiquare "which is made of A C is equall lento the fiquare " which is made ofC E. Wherefore the fquares "Which are made ofC A and C E are double to the fiquare "which is made of A C. Eut (by the 47* of the firft) the jquare "Which is made of E A is equallto the fquares ' which are made of AC and C E ( For the angle ACEis a right aw gle) "Wherefore the fiquare of A E is double to the fiquare of A C. Agayne forafi much as E Gfis equalllmto G F, the fiquare therfore "Which is made ofE G is e* qual to the fiquare "Which is made ofiGF. Wherfore the fquares "Which are made ofGE and G F are double to the fiquare "Which is made of G F. Eut ( by the 47* oj the firU) the fiquare "Which is made ofiEF is equallto the fquares "Which are made of EG and G F. Wherfore the fiquare "Which is made ofE F is double to the fiquare "which is made ofiGF. Eut G F is e quail lonto C ID. Wherefore the fiquare "Which is made of EF is double to the fiquare " which is made of C E) . And the fiquare "Whiche is made of A E is double to the fiquare "Which is made of A C. Wherefore the fquares ' which are made of A E and E F are double toy fquares which are made of AC and CE). Eut (by the 47* of the firH)the fiquare which is made of AFis equal to the fquares which are made of A E and EF( Forj angle A E F is a right angle ) . Wherfore the fiquare "Which is made of AF is double to the fquares "Which are made of AC iyC Ed.Eut ( by the 47* o f the firfl)y fquares "Which are made ofiAE) and E) F are e* quail toy Jquare "Which is made of A F.Fory angle oty point E> is a right angle . WJ:>erfiore the fquares " which are made ofiAE) and E) F are double toy fquares "Which are made of AC and C E). Eut E) F is equall Ipnto E) E. Wherfore the fquares "which are made of A E) and E)Ej are double to the fquares "Which are made of AC and C E). If therfore a right line be deuided into two equall partes and into two lane quail partes jbe fquares "Which are made of the lane quail partes of the "whole yare double to the fquares "which are made of the halfe lyne3 and of that lyne "Which is betwene the fie (lions: * which "Was required to he proued. E f An example of this propofition in numbers. Take any enen number as 1 2. And deuide it hrft equally as into <?. and 6t &then m» equally as into 8 ,& 4. And take the difference of the halfe to one of the vnequal partes V.ij» which The fecondHooke which is 2 .And take the fquare numbers of the vnequalJ partes 8, and 4,which are and itf.’and.adde themtogether,which make 80, Then take the fquares ofthehalfe 6, and of the differece 2: which are 3 d^anciq: which added together make 40.Vn.to which numberthe number compofed of the fquares of the vnequall partes,whiche is So, i# double. As you fee in the example* The whole. Multiplication of eche vnequalpart into himfelf. >< Multiplication of the half and of the difference eche into himfelfe. 12 8 8 4 '4 6 6 ii 2 2 ^ J* the equal! pat tes 8 4 J 2 the vnequall partes 64 16 64 16 80 3 6 4 40 the difference ofthc halfe to one of the partes r the number compofedof the C fquares of the vnequal partes ' > double to the number compofed of the fquares of the halfe,and of the difference. The demonftration wherof followeth in Barlaam*. J The ninth Tropo/ition. If a number be deuided into tVoo e quail numbers, and againe be deuided into two inequall partes: the fquare numbers of the vnequall numbers, are double to the fquare which is made of the multiplicand on of the halfe number into it felfe, together With the fquare Whiche is made of the number ft bed tWene them. For let the number zA B being an cuen number be deuided into two cquall numbers A C & C B: Sc into two vnequall nubers AD and D A.Then / fay that the fq uare num¬ bers o (AD and D B,txt double to the fquares which are made of the multiplication of the numbers AC and CD into themfelues. For forafmuch as the number si B is an euen number, and is deuided alfo into two equal numbers A C and Ci^and afterward into two vnequal nubers A D and DB-.t herefore the fuperficial nuber produced of the multiplicand of the nubers AD &DB> thone into the other, together with the fquare of the number D C,is equal to the fquare of the number ACfby the fife proportion ) Wherfore the fuperficiall number produced of the multiplication of the numbers AD and D B the one into the other twife, together with two fquaresof the number CD}is double to the fquare of the number zA C, Forafmuch as alfo the number A B is deui¬ ded into two equal numbers zA Cand C3,therfore the fquare number of AB is qua¬ druple to the fquare number produced of the multiplication of the number <sA C into it felfef by the q.propofition) .MOfeouer forafmuch as the fuperficiall number produ¬ ced of the multiplication of thenumbers A the one into the other twife to¬ gether, with two fquares of the number D C is double to the fquare number of CA.&c forafmuch of Euclides Element? s . Fol.j 5< forafmuch as there are two numbers* of whiche the one is quadruple to one and the felfe fame number,and the other is double to the fame hum ber: therefore that number whiche is quadruple fhall be double to that number whiche is double- Wherefore the fquare of the number A B is double to the number produced ofthe multiplicand of the numbers ~ jD and D B the oneintotheothertwife together with the two fquares of the number!) C.Wherfore the number which is produced of the mul¬ tiplication ofthe numbers A D and D B the one into the other twife* is lelfe the halfe of the fquare ofthe number A B by the two fquares ofthe ^ numbers D C. And forafmuch as the nuber produced of the multiplica¬ tion ofthe nuber s AD 8c D B the one into the other tw,ife, together with the nuber copofed of the fquares ofthe numbers A D an&T) B is(by the 4.propofiaori)equall to the fquare ofthe number.^ A ; therforethe nu¬ ber compofed of the fquares of the numbers A D 8c D "Bis greater then the halfe ofthe fquare nuber ofc^ B, by the two fquares of the number I) C. And the fquare ofthe numbered/ 2? is quadruple to the fquare of 6 the number A C.Wherfore the number compofed of the fquares of the numbers A D and® B is greater then the double of the fquare of the number ACby two fquares of the number D C.Wherfore the faid num¬ ber is double to the fquares ofthe numbers A C and C‘Z>. 7f therefore a number be deuided &c,which was requiredto be demonftrated. ,'D 8 8 2 2 <5q. the fquare of the vnequall part AD 4 the fquare of the vncquall part £ JD 5 5 2 5 the fquare of the halfe A C* 9 the fquare of C D3nameiy,of the number fet betwene. 9 3 4 the fquares of the halfc*and of the number fet bctwenc. *4 4 68 34 I2* ~ 68 the fquares of the vncquall partes. . • ■ t- - - . *. ■ - ^ 1 _ _ ■ _ * * 1 « .* ' * • - -4 The 10. Theorems. The xo.Trofofition . If a right line be deuided into two equal partes vnto it be added an other right line direBlyithe fquare which is made of the whole & that which is added as of one line , together with The fecond'Booke the fquare whiche is made of the lyne whiche is added \ thefe mo fcjuares ( 1 fay') are double to thefe fquarcsynamelyjo the Jquare which is made of the halfe line to the fquare which is made of the other halfe lyne and that whiche is added > as of one lyne . ConJtrntlim. Dtmtnflra- tion . V ppofe that a cert ay ne right line ABbe deuided into tleo equal l partes \in the point C.And Vnto it let there he added an other right line dir eft* ^^lyjiamel0d D. The I fayjhat the fquar.es which are made of the lines *A D and D B are double to the fquares Which are made of the lines _A Cand C D. (Rayfeyp (by the n, of the firH ) from the point C^nto the right line ACT) a perpendiculer lyneyand let the fame be C E. And let C E (by the 5, of the fir ft) be made equallvnto either of thefe lines J Cand CB. And (by the firft petiti * on) draw right lines from E to A^andfrom E to B. And(by they, of the fir ft) by the point E fir aw a line parallel Vnto C Dymd let the fame be E EAnd(byy felfjame )by the point T) draft; a line parallel Vnto C E and let the fame be IDE. Andforafmuch as Vpon thefe parallel lines CEzs'DE ligbtetb a certain right line E Fyherf m( by the zyjfthe firft) the angles CEFandEFD are equal Vnto tft>o right angles jETher fore th e angles FE B^and EFD are lefle then tit 0 right angle sJBut lines produced from angles leffe then two right angles( by the fifth peticion)at the length meete together. VTberfore the lines EB andF D beyng produced on that fide that the line B Distill at the length meete to * gether: Produce them and let them meete together in the point G . And (by the firft peticion)draW a line from A to G, Andforafmuch as the line A C is equal l Vnto the line C Ey the angle alfoAECis (by the of the E F firfl)equall ’Vnto the angle E A Ct And the angle at y point Cis a right angle. W her fore eche of thefe angles E A C<& and A EC is the halfe of a right angle. And by the fame reafon eche of thefe angles C E B, and ETC is the halfe of a right an fie. Wherefore the angle A E Bis a right angle. And fotafmuch as the angle ETC is the halfe of a right angle fherfore (by the 1$. of the firH)tbe angle DBG is the half of a right angle. But y angle BDGis a right angle(for it is equal Vnto the angle D CEfor they are alternate ahgles)Wher fore the angle remaining DG B is the halfe of a right angle. VV her fore (by the 6 common fentence of thefirfl)the an * gleDGB is equall to the angle DBG # VF her fore (by the 6 .of the firft) the fide m is equall Imto the fide GD. Agayne forafmuch as the angle E G F it thi balfeofci right angle : and the angle at thepointe F is a right angle: for (by the 3Fof the first) it is equall lento the oppofte angle EC ID. Wherefore the angle remay ning F EG is the halfe of a right angle. Whet fore the angle E G F is e* quail to the angle F E G. Wherfore ( by the 6. of the frit the fide FF Eis equall lento the fide F G. And forafmuch as EC is equall Imto C A, the fquare dfe i! Ariel: is made ofE C is equall to the fquare which is made of C A* Wherefore the fquares which are made of CEand C A are double to the fquare "Which is made of AC. Eut the fquare "Which is made ofE A is (by the 47* of the fir A) e* quail Imto the fquares which are made of EC and C A. Wherefore the fquare Which is made of E A is double to the fquare which is made of A C. Again e for* afmuch as G F is equall Imto E F/he fquare alfo which is made ofGF is equall to the fquare winch is made ofFE. VVherfore the fquares which are made of G F andE F are double to the fquare which is made ofE F. Eut (by the 47 •of the fir A) the fquare which is made of EG is equall to the fquares which are made of GF andE F. Wherefore the fquare which is made of EG is double to the fquare Which is made ofE F.EutE Fis equall Imto C D, wher- forey fquare which is made ofE G is double to the fquare Which is made of C D. And it is proued the fquare which is made of E A is double to the fquare which is made of A C. Wherfore thefquar es Which are made of A E andE G are double to the fquares which are made of A C and C D. Eut(by the 47- of ths fir A) the fquare which is made of AG is equall to the fquares which are made of A E and E G, Wherefore the fquare which is made of AG is double to the fquares which are made of AC and C ID. Eut Imto the fquare which e is made of AG are equall the fquares which are made of A D and D G. Wherfore the fquares which are made of A D and D G are double to the fquares which are made of A C andD C • Eut D G is equall Imto D E.Wherfore the fquares which are made of A D And D E are double to the fquares which are made of AC and D C. Iftherfore aright line be deuidedinto two equall partes y and Imto it be added an other dyne directly , the fquare which is made of the Whole and that which is added , as of one line together With the fquare which is made of the line which is added ghefe two fquares (I fay) are double to thefe fquares y namely } to the fquare which is made of the halfe lyne, and to the fquare which is made of the other halfe lyne and that which is added gs of one lyne: which Was required to be proued. v • . ... • - •- . . •' . f An other demonAration after Eelitarius. V.iili. Suppofe The fecondBooke Suppofe that the lyne AB be deuided into two equal! partes in the poytiteC. And vnto it let there be added an other right lyne dire&ly, namely, B D. Then I fay that the fquare of AD together with the fquare of B D is double to the fquares of A Vpon the whole line AD defcribe a fquare AD E F.And ypon the halfe Ivne A Cde-? fcnbe the fquare A C G H.And produce the Tides G H and C H till they cut' the lides E F & D F,wherby ilialbc defcribed the figure H L K F, which ihalbethe fquare of the line C D the Corollary of the 4. of this boke,& by the 34.Propofition of the i.)itis manifcfhf we draw the diameter C D. For the lyne K F is equall to the line C D. And making alfo the lines H M and H N equall to either of thefe lynes A C and C B, drawe the lynes M G and NP cutting the one the other right angled wife in the point' Q. Ei¬ ther of which lynes let cutthefides of the fquare A D; E Fin the pointes OandP.Nowitnedeth nottoproue that the figure H Qis the iquare of the lyne A C,fcyng that it is the fquare of the line C B : as the figure QJ is the fquare ofthe line BD ; neither alfo needethit to proue that the paralleiograme H P is eq uall to either of the fupplementes E H and H D : nor that the fupplc- mentcs N O and QJ. are equall. For all this is manifeft cue by the forme of the figure, for that all theangles a- bou t the diameter arc half right angles, & the fidcs are equall. Wherfore if we diligently marke of what partes the fquare HF which is the fquare of CD, iscompo- fed,we may thus reafo. Forafmuch as the whole fquare E D is compofed of the two fquares A H and H F and of the two fupplementes E H and H P,we muftproue that thefe fupplementes with the fquare QJJwhich is the fquare of the line B D) are equall to the two fquares A H and H F.For then (hall we proue that thefe two fquares AH 6£ HF taken twile are equall to the whole fquare DE together with the fquare of QF9 which thing we tookefirftin hand to proue. And thus do I proue it. The Supplement E H is equall to the paralleiograme H P. And the fquare A H to¬ gether with the lefier fupplemet, N 0,is equall to the other fupplemet H D (by the firft common fentence fo oftentymes repeted as is neede ) wherfore the two fupplementes E H and H D are equall to the fquare A H and to the Gnomon K H L P QjO. If therfore vnto either of them be added the fquare QJ : the two fupplementes E H and H D to¬ gether with the fquare of QJ dial be equal to the fquare A H, & to the Gnomon K H L P QO and to the fquare QJJBut thefe three figures do make the two fquares A H and H F.VVherforc the two fupplementes E H and H D together with the fquare QJF arc e- quall to the two fquares AH and H F, which was the fecond thing to be proued. Wher¬ fore the two fquares A H and H F beyng taken twife are equall to the whole fquare D E together with the fquare of QJ3. Wherfore the fquare D E together with the fquare QJ is double to the fquares A H and H F: which was required to be proued. f jin example of this Tropofition in numbers. Take any euen number as 1 8: and take the halfe of it which is 9. and vnto 18. the whole, adde any other number as 3 .which maketh 2 1 . Take the fquare number of 1 1 . (the whole number and the number added) which maketh 441. Take alfo the fquare of 3 C the number added) which is p. which two fquares added together make 450. Then adde the halfe number p. to the number added 3. which maketh 12. And take the fquare ofp.the halfe number and of 1 2, the halfe number and the number added which fquares are 8 1. and 144. and which two iquares alfo added together make 235; ■vnto which fumme the forefayd number 450, is double. As you fee in the example. The The whole. The number added. ententes • r 18 Fol.yj. Multiplication of the whole and the number addedinco himfelf. Multiplication of the number addeddnto himfelfe. Multiplication of the halfe in¬ to himfelfe. Multiplication of the halfe, & the number added into it felfe. >■«< l 21 21 21 21 42 44I 44I 3 9 . - 9 9 81 12 1 2 24 12 144 1 the halfe- 81 iH 225 ~ The number compofed of the fquare of the whole & the number added and of the fquare of the number added, -doublet© the number compofed of the fquare of the halfe and of the fquare of the halfe and the number added. The demonftration wheroffolloweth in Barlaam* r. - The tenth (Proportion, If ten tnmnomber be deuided into two squall nombers , and vnto it be added any other nomber: the fquare nomber of t he Whole nomber compofed of the nober and of that which is added , and the fquare nomber of the nober added.-thefe tWo fquare nobers( I fay )added together , are double t@ thefe fquare nombers, namely, to the fquare of the halfe nomber t and to the fquare of the nomber compofed of the halfe nomber and of the nomber added. Suppofe. that the nomber c A B being an euen nomber be deuided into two cquall nombers AC andCZ? : and vnto it let be added an other nomber BD . Then I fay, that the fquare nombers of the nombers AD an dDB are double to the fquare nombers of *XC and CD. For forafmuch as the nomber AD is deuided into the nombers AS and BT): therefore the fquare nombers of the nombers AD and DB are equall to the fu~ perficiail nomber produced of the multiplication of the nombers AD andD2? the on into the other twife,together with the fquare ofthe nomber .^.5 f by the 7 propofitio) But the fquare of the nomber^ is equal tofowerfquaresof either ofthe nombers ACotCB (for ^4Cis equall to the nomber CB ): wherforealfo the fquares ofthe nom¬ bers AD an dDB are equall to the fuperficiall nomber produced ofthe multiplication of the nombers AD and^JDi? the one into the other twife, and to fower fquares ofthe nomber BCox CA, And forafmuch as the fuperficiall nomber produced ofthe multi¬ plication of the nombers AD and DB the one into the other, together with the fquare ofthe nomber Ci?, isequal to {quareofthe nomber CD(by the 6 propofitio) : therfore the nomber produced of the multiplication of the nombers AD and DB the one into the other twife together with two fquares of the nomber CB,is equall to two fquares ofthe nomber CD . Wherefore the fquares of the nombers AD and DB are equall to X-£» two rJL The fecondTHookf two fquares ofthe nomber C ® , and to two fquares of the nomber AC . Where>- fore they are double to the fquares ofthe numbers AC and C®. And the fquare of the nomber <s A D is the fquare of the whole and of the nomber added ; And the fquare of ® B is the fquare of themombe r added : the fquare alfo of the nomber CD is the fquare of the nomber compofed of the halfe and of the nomber added : If there¬ fore an euen nomber be deuided.&c. Which was required to be proued. 8 8 2 2 the fquare of A D 4 the fquare of® B the fquare of C ®,namely.,of the number compofed of the halfe and of the number added. 3 ay Vd-f"1 J 9 the fquare ofthe halfe A C. 2$ 9 34 k 68 J ■ rs8| J 341 2. A. yk^Tbe I, Trobleme% TheufPropoftion . To deuide a right line geuen in fuch fort, that the reBangU fgure comprehended ynder the whole >and one ofthe partes , /hall he e quail vnto the fquare made ofthe other part . fppoje that the right line geuen he A <B. "Flow it is required to deuide || the line A <B in fuch forty that the re Bangle figure contajned Tmder the vhole and one of the partes yfball be equal l Ivnto the fquare “Which is made of the other part.lDefcribe (by the 46. of the fir ji) 'Vpon A 35 Conttrutlion. a fquare ABCftD. And ( by the 10. of the firjl ) deuide the line A C into two equall partes in the point E,and draw a line from B to E. And (by the fecond petition)extend C A lento the point F .And (by the 3. of the firfi)put the line E F equall Tmtoj line B E. And (by the 4 6. of the firft)vpon the line A F defenbe a fquare FG AH. And (by the 2. petition) extend GH Imto the point Ff T hen I fay that the line yfB is deuided in the point H in fuch fort, that the reRangle figure -which is compreheded Tender A B and B FI is equall to the fquare -which is made of Demon ft ratio AH. For forafmuch as the right line A C is deuided into two equall partes in the poynt E,and 'Vnto it is added an other right line jK G _p % H A E V K C of Euclides Element es . Fol.jS . ^F. Therefore (by the 6. of the fecond) the re&angle figure contayned Wider C F and F A together loith the fijuare 1 vhich is made of A E is equall toy fquare lohichis made ofR F . But E F is equall Wito E B . VVherefore the re Bangle figure contayned lender C Fynd F A together loith the fquare lohich is made of E A is equall to the fquare lohich is made of EE. 'But ( by 47. of the firfi ) Wito the fquare "Which is made of EB are equall the fquares lohich are made of B AandAE . For the angle at the poynt A is a right angle . VVherefore that lohich is contayned "wider C F and F A} together loith the fquare lohich is made of A E, is equall to thefquares lohich are made of BA and yfE . T ake away the fquare lohich is made of A E lohich is common , to them both : VVherfore the reElangle figure remayning contayned Wider CF and F A is equall Wito the fquare lohich is made ofAB . And that lohich is contained Wider the lines C F and F A is the figure F If . For the line F A is equall Unto the line F G. And the fquare lohich is made of A B is the figure A ID ., VVherefore thefi * gure F if is e quail Wit 0 thefigure AD . Takeaway the figure A if lohich is common to them both. VVherefore the reftdue> namely } thefigure FH is equall Unto the rejidue ynamely}Unto thefigure Elf) . But the figure HAD is that, lohich is contayned Wider the lines AB and BH,for AB is equation* to BD . And thefigure F His the fquare lohich is made of AH. VVherfore the re&angle figure comprehended "Under the lines AB and BH is equall to the fquare lohich is made of the line H A . VVherefore the right line geuen AB is deuidedin the point H, in fuel? fort that the re&angle figure contayned Under AB and B H is equall to the fquare lohich is made of AH; lohich loas required to be done. ' Thys propofition hath many fingular vfes. Vpon it dependeth the demonftration Many and of that worthy Probleme the 10. Propofition of the 4.booke : which teacheth to de- fingultr vfes feribe an Ifofceles triangle, in which eyther of the angles at the bafe (hall be double to of this props- the angle at the toppe . Many and diuers vfes ofa line fodeuided Hull yon finde in the fiiion. ij.booke of £uchde. Thys is to be noted that thys Propofition can not as the former Propofitions rI . of thys fe'cond booke be reduced vnto numbers . For the line EB hathvnto the ■ fine AE no proportion that can be named, and therefore it can not be exprefled VenducedU by numbers. For forafmuch as the fquare qf EB is equall to the two fquaresof to numbers * AB and AE (by the 47.ofthe firfi) and AE is the halfe of A B, therefore the line BE is irrationall . Foreuenas two equall fquare numbers ioyned together can not make a fquare number : fo alfo two fquare numbers, of which the one is the fquare of the halfe roote of the other, can not make a fquare number . As by an example. Take the fquare of 8. which is 64. which doubled, that is, 128. mi keth not a fquare number • So take the halfe of 8 . which is 4* And the fquares of 8 . and 4. which are 64. and 1 6. added together Hkewyfe make not a fquare num¬ ber . For they make 8 o. who hath no roote fquare . Which thyng muft ofnecelii* tic be if thys Probleme fliould haue place in numbers. But in Irrationall numbers it is true3 and may by thys example be declared. X.ii. Lee r Let 8,be fo deuided,that that which is produced of the whole into one of his partes Ihall be equall to the fquare number produced of the other part. Multiply 8. into him felfe and there (hall be produced 64. that is, the fquare CD.Deuidc 8. into two equall partes, that is,into 4, and 4. as the line E oxEC . And multiply 4. into hym felfe,and there is produced 1 <5, which adde vnto 64, and therefhall be produced 80: whoferooteis Vg^ 80: which is the line £ 2? or the line E F by the 47, of the firft* And forafmuch as the line EFisdif 80, & the lyne E A\s 4* therfore the lynctAt Fis 80— 4,And fo much (hall the line AH be. And the line B H (hall be 8 — */§"* 80— 4,that is j 1 2 — Vfr 80. Now the 12 — 8 o multiplied into 8 dial be as much as 8 o — 4, multiplied into it felfe. For of either of them is produced p 6 — y/ 5120. he uffheorme . The izfPropofition . In ohtufeangle triangles, the fquare which is made of the fide fubtending the obtufe angle, is greater then the fquare s which are made ofthefides which comprehend the obtufe angle , by the reB angle figure, which is comprehended twife vnder one ofthofe ftdes which are about the obtufe angle ,ypon which being producedfalleth a perpendicular line 3 and that which is outwardly taken betwene the perpendicular line and the obtufe angle. Demonstra¬ tion. $ V^ppofe that ABC he an ohtufeangle triangle bauing the angle B AC obtufe ,and from the point B(by the 12* of the firft)dr aw a perpendicular line lonto CA produced and let the fame be BID. Then I fay that the fquare 1 vhich is made of the fide B C3 is greater then the fquares "Which are made of the fides B A and A C,hy the re hi am _ _ fff gle figure comprehended lender the lines CA and AD twife . For forafmuch as the right line CD is by chaunce deuided in the poynt A } therefore (by the 4. of the fccond ) the fquare "Which is made of CD is equall to the fquares "Which are made of C A and A D,and lonto the reflangk figure contayned Wilder CA and AD twife. But the fquare "Which is made of DB com * mon Ivnto them both . Wherefore the fquares -which are made of CD and D B are equall to the fquares ' which are made of the lines C A, A Dj and D Byand nto the red angle figure contayned lender the lines CA and AD twife .But (by the the firflfihe fquare yohich is made of CB is equall to the fquares lehtch are made of the lines • ' if y. ■ ■M ofEuclides Elementes . Foi 77* CD and /D/8. For the angle at the point D is a right angle . AndWnto the fquares lohich are made of AID and D H (by the felfe fame ) is equall the fquare t/hich is made of A /8 . VVherfore thefquare H vhich is made of CHys e- quail to the fquares which are made of CA and A <B and "Onto the reElanglef gure contayned loader the lines C A and A ID twife. Wherfore j fquare 1 vhich is made of C/S fits greater then the fquares which are made of CA and AD by the rectangle figure contayned Tender the lines CA and AD twife. In obtufe- angle triangles therefore yhe fquare which is made of the fide fiubtending the ob * tufe angle ,is greater then the fquares which are made of the fides YOhich com * prehend the obtufe angle Jay the reBangle figure VOhich is comprehended twife louder one of thofe fides which are about the obtufe angle gopon which being pro* duced falleih a perpendiculer lyne^and that which is outwardly taken betwene the perpendiculer lyne and the obtufe angle : which was required to be demons Jirated. Of what force thys Propofition, and the Propofition following, touching the meafuring of the obtufeangle triangle and the acuteangle triangle, with the ayde of the 47. Propofition of the foil booke touching the rightangle triangle, he fhall well perceaue,which fiiall at any time neede the arte of triangles in which by thre thinges knowen is euer iearched out three other thinges vnknowen,by helpeof the table of arkes and cordes. The n.Tbeoreme. fhc 13 fPropofition. 5^ h acuteangle triangles 3the fquare which is made of the fide that fubtendeth the acute angle js lejfe then the fquares which are made of the fides which comprehend the acute an¬ gle >b the reBangle figure which is coprehended twife vnder one of thofe fides which are about the acuteangle , vpo which fillet h a perpendiculer lyne , and that which is inwardly ta~ ken betwene the perpendiculer lyne and the acute angle . ■ Vpp°fe that ADC be an acuteangle triangle ha* the angle aty point D acute ys( by the iz.of L_ns! t hefirU from the point A dr aw "Onto the lyne D C a perpendiculer lyne JD. Thenlfaythat thefquare lohich is made of the lyne JC is lefie then the fquares ^hich are made of the lyne CDandD Ajby the reBangle figure conteyned bonder the lines CD and HD twife. For forafmuch as the right lyne <B C is by chaunce deuided in the point Dyher fore (by the 7. of the fiecond) the fiq uares X.iij. which A DemnJlratiS The fecond Boothe lohich are made of the lines C B and B D are e quail to the reBangle figure com tamed lander the lines C B a?id D B twife and Tanto the fquare lohiche ismade of line C D .Tut the fquare lohich ismade of the line !D A common lanto them both . VVherfore the fquares lohich are made of the lines CB , BD,and(D jf3are equall lanto the re El angle figure contayned lander the lines C B and B ID twife , and lanto the fquares lohich are made of AD and D C. But to the fquares lohiche are made of the lines B D and D A is equal y fquare lohich is made of the line JIB : for tl/ angle aty point D is a right angle . Andlanto the jquares lohiche are made of the lines A D and D C is equall the fquare lohiche is made of the line A C( by the a-.ofy firfi):loberfore the fquares lohich are made of the lines C B and B A are equal to the fquare lohich is made of the line A C,and to that lohich is contain 7ied louder the lines C Band BD twife . Wherfiorethe b fquare lohich is made of the line A C beyng taken alone js lejfe then the fquares lohich are made of the lines C B and BA by the rectangle figure , lohich is con * tainedlonder the lines C B and B D twife . In rectangle triangles therfore the fquare lohich is made of the fide that fubtendeth the acute angle, is lejfe then the Jquares lohich are made of the fides lohich comprehend the acute angle , by the rectangle figure lohich is comprehended twife louder one of thofie fides lohich are about the acute angle, lopon lohich falleth a perpendicular line , and that lohich is inwardly taken betwene the perpendicular line and the acute angklehich loas required to be proued. 1] jt Corollary added by Orontius. A Corollary. This Propo/i- tion true in all kindes of triangles * Hereby is eafily gathered,that fuch a perpendicular line in redangie triangles falleth ofneceffitie vpon the fide of the triangle, that is, neyther within the trian¬ gle, nor without. But in obtufeangle triangles it falleth without, and in acuteangle triangles within . For the perpendicular line in obtufeangle triangles, and acute- angle triangles can not exa&ly agree with the fide of the triangle : for then an ob- tufe & an acuteangle ihould be equal to a right angle,contrary to the eleuenth and twelfth definitions of the firft booke . Likewife in obtufeangle triangles it can not fall within, nor in acuteangle triangles without: for then the outward angle of a triangle fhould be lefte then the inward and oppofite angle,whichis contrary to the i ^.of the firft. And this is to be noted, that although properly an acuteangle triangle, by the definition therof geue in the firft booke,be that triangle, whofe angles be all acute: yet forafinuch as there is no triangle,but that it hath an acute angle,this propofith on is to be vnderftanded,& is true generally in all kindes of triangles whatfoeuers and may be declared by them, as you may eafily proue. The of Euclides Elements s. FoL 80, The lEProbleme. Thev^.Tropofition. Vnto a reBiline figure geuen3to make a jquare squall. ^The ende of the fecond Booke ofEuclides Elementes. Cofijiruftm. Dmo»tJlraii$ The Argument of this kooks* The fir ft defi¬ nition* fflby circlet take their equality of their diame- ten or femi¬ diameters • i[.The third books of Eu- elides Elementes. His third booke ofEuclide entreateth of the moft perfed: figure, which is a circle . Where¬ fore it is much more to be eftemed then the two bookes goyng before , in which he did fet forth the moft fimple proprieties of rightlined figures . For fciences take their dignities of the worthynes of the matter that they entreat of.But of al figures the circle is of moft abfoluteperfedion,whole proprieties and pafsions are here fet forth,and moft certainely demo- ftrated.Here alfo is entreated ofrightlines fubten- ded to arke's in circles : alfo of angles fet both at the circumference and at the centre of a circle,andofthevarietie and differences of them.Wherfore the readyng of this booke , is very profitable to the attayning to the knowledge of chordes and arkes.lt teacheth moreouer which are circles con- tinget,and which are cutting the one the other : and alfo that the angle of contin- gence is the leaft of all acute rightlined angles:and that the diameter in a circle is the longeft line that can be drawen in a circle . Farther in it may we learne how, three pointes beyng geuen how foeuer(fo that they be not fet in a right line), may be drawen a circle palling by them all three, Agayne, how in a lolide body , as in a Sphere,Cube,orfuchlyke,maybe found the two oppofite pointes . Whiche is a thyngvery necefiaryand commodious : chiefly for thofe that fliall make inftru- mentes feruyng to Aftronomy,and other artes. ! Definitions . Equal! circles are fuch^hofe diameters are equally or who/e lynes drawen from the centres are equal! . The circles A and B are equal, if theyr diameters,namely,E F and C D be equaShor if their femidiameters , whiche are lynes drawen from the center to the circumference^ namely A F and B D be equall. i The reafon why circles take theyr equalitie , of the e- qualitie of their diameters or femidiameters is , for that a circle is delcribed by one re- uolution or turnyng about of the lemidiameter, hauing one of his endes fixed . As if you Imagine the lyne A E to haue his one point namely A faftened,and the other end namely E to it o/Sudides Elementes. Fol.2u it come to the place where itbega to moue , it fhal fully defcribe the whole circle. Wherefore ifthefemidiameters bee equall5the circles of neceffity e muft alfo be equal!: and alfo the diameters. By thys alfo is kno wen the definition of vnequall circles. Definition of imeqHtUw* Circlestyhofe diameters orfmidiameters are vneqnatt, are alfo vneqnai . *And that circle Wft fybicb hath the greater diameter or femidiametcr, is the greater circle » and that emit Which hath the lejfe diameter or femidiamet ert it the leffe circle „ As the circle t M is greater then the circle I K , for that the diameter L M is greater then the diameter I K: or for that the femi- diameterG Lis greater then the femidiameterHI. right line is fay d to touch a circle jtohich touching the cir > ele and being produced cutteth it not . mm* As the right lyneEFdrawen from the point E 9andpa%ng by a point of the circle* namely ,by the point G to the point .F on¬ ly toucheth the circle G H,and cutteth it not, nor entreth within it.For a right line entryng within a circle, cutteth anddeui- deth the circle . As the right lyne K L de¬ ni deth and cutteth the circle K L M , and entreth within it : and therfore toucheth it in two places . But a right lync tou- chyng a circle, which is commonly called a cotingent lyne,toucheth the circle one* ly in one point. df contigent tine* % Circles are fayd to touch the one the other ythicb touching the V’M&foh one the other jut not the one the other* ***** As the two circles AB and BC touch the one the other ♦ For theyr circumferences touch together in the poynt B * But neither of them cutteth or deuideth, the other . Neither doth any part of the one enter within theo« ther.And fuch a touch of circles is euer in one poynt onely: yhich poynt oncly is common to them both .As the poynt B is fit the confc* tehee of the circle A B,and aUb is io, the circu- ifereace of the circle BC. The touch of emits is tuev in one point mdj. , Aa-f. Circles Circles mg South toge¬ ther two md- mrofurdjes. Fourth defi- nuion. Fife defini¬ tion. touch together two matier of wayes , either outwai^ly &£qij,^ wholy without the other : or els the one being cojitayne4 withiuthe otfc. As the circles D E and D F : of which the one D E contay- neth the other , namely D F : and touch the one the other in the poynt Z>;and that oncly poynt is common to them both : neitherdoththeone enter into the other »Ifanypart ofthe pne enter into any, part of the other,then the one cutteth and feideth the other , and toucheth th? one the other not in one poynt oneiy as in the other before , blit in two pbintes, and haue alfo a fupcrficies common to them both. As the cir¬ cles G H ifand HL K cut the one the other in two poyntes H and X;and the one entreth into the other : Al¬ fo the fuperficies AfiC is common to them both; For it is a part of the circle G H K , and alfo it is a part pf the circle HL K. %ight lines in a circle are fay d to be equally diBant from the cen~ freshen perpendicular lines drawen from the centre ynto thoje lines are equall, <*And that lint is fayd to be more di~ Hantfvpon wbomfalleth the greater perpendicular line. As in the circle tABCDw hofe centre is E, the two lynes JI B and CD haue equall difiance from the centre E .* bycaufe that the Iyne EF drawen from the centre E perpendicularly vpon the lyne A 2?,and the lyne E G drawen likewife perpendi- larly from the centre E vpon the lyne CD are equall the one to theother . Butin the citdzH KLM whofc centreis 2\^the lyne H if hath greater diftance from the centre 2^ then hath the lyne L M : for that the lyne O 2^drawen from the centre ^perpendicularly vppon the lyne HKis greater then the lyne T^JF which is drawen fro the centre perpendicularly vpon the lyne L So likewife in the other figure the lynes AB and D Cin the circled BCD are equidiftant from the centre C,bycaufe the iyncsOCandiy jP perpendicularly drawen from the centre G vppon the fayd lynes A B and D Care equall . And the lyne A B hath greater diftance from the centre G then hath the the lyne EF , bycaufethclync O G perpendicularly drawen from the centre G to the lyne cAB is greater then the lyne H G whiche is perpendicularly drawen from the centre G to the lyneEF, A feBion or figment of a circle 9 is a figure coprehendedmder a right line and a portion ofthe circumference of a circle. As the As the figure ABCis a fedtion of a -circle bycaufe itis comprehended vnder the right lyne AC and the circumference of a circle A. B C . Likewife the figure D E Ais a fedtion of a circle , for that it is comprehended vnder the right Ivne D F , and the circuference D E F. And the figure A BC for that it cotaineth within it the centre of the circle is called the greater fection of a circle : and the figu re CD E Fis the lefle fection of a itis w holy without the centre of the circle as it was noted in the 1 6 . Definition ortne firft boofce* . .. f - *tAn angle ofafeBion or fegment ? is that angle which is con < tajneciyncier a right line and the circuference of the circle . As the angle A B C in the fedtion A B C is an angle of a fee- tiort , bycaufe it is c6ntained of the circumference B A C and the right lyne B C . Likewife the angle C B D is an angle of the ^ fedtion B D C bycaufe it is contayned vnder the circumference B D C,andthe right JyneB C . And thefe angles are commonly I called mixte angles, bycaufe they are contayned vnder a right ' lyne and a crooked . And thefe portions of circumferences are commonly called arkes, and the right lynes arc called chordes, or right lynes fubtended. And the greater fedtion hath euer the greater angle,and theleffe fedtion the Idle angle, Jn angle is fayd to be in afeBionjtohein the circumference is takgn any poynt^andfrom thatpqynt are drawen right lines to the endes of the right line which is the bafe of the fegment ? the angle which is contayned vnder the right lines drawen from the pojnt, is ( Ifayfaydto be an angle in a JeBion . As the angle A BCfs an anglein the fedtion A B C , bycaufe from the poy nt B beyng a poynt in the circumference A B C are drawen two right lynes B C and B A to the endes of the lyne A G which is the bafe of the fedtion A B C . Likewife the angle ADC is an angle in the fedtion A D C, bycaufe from the poyrit D beyng in the circuference A D C are drawen two right lynes,namelv,D C & D A to the endes of the righ t line A C which is alfo the bafe to the fayd fedtion A D C-So you fee,it is not all one to fay, an am gle of a fedtion,and an angle in a fedtion.An angle of a fedtion co- fifteth of the touch of a right lyne and a crooked. And an angle in a fedtionis placed on the circumference , and is contayned of two right lynes . Alfo the greater fedtion hath in it the leifc angle , and the leife fedtion hath in it the greater angle.- 'v' ■ o-.' . " '‘But when the right lines which comprehend the angle do re - ceaue any circumference of a circle jben that angle is fayd to . he correfgondent^and to pertaine to that circumference* Aa.i/, As the Sixt defm~ tion. Mfot Angles, Arkes. C hordes , Seuenth finitim «•. Difference of an angle of £ Seffion.and of an an Ae in a Semon» Eight deft nition » plinth dtfi- mtion . Tenth dcfini - tion. T wo defini¬ tions, Brft, Semi* As the right lynes B A ~ €,and reeeaue the circumference A D C iherforc the angle A B C is fayd to fubtend and to pettaine to the circuference ADC. And if the right lynes whiche. caufe the angle, concurrein the centre of a circle : then the angle is fayd to be in the centre of a circle « As the angle E F D is fayd to be in the centre of a circle, for that it is comprehended of two right lynes F E and F Ds rhiehe concurre and touch in the centre F. And this angle likewife fubtendeth the circumference EG D : whiche circumference alfo, is the meafure of the greatnes of the angle E F D* ' of a circle is (an angle being fetal the e centreof a circle ) a figure contayned ynder the right lines which make that angle 9and the part of the circumference re* ceaucd ofthem. As the figure A B C is a fedor of a circle, for that it hath an angle at the centre,namely the angle B A C,& is cotained of the two right j lynes A B and A C ( whiche contayne that angle and thedreumfe. rence receaued by them. Likefegmentes orfellions of a circle are thofe9 which ham equall angles %or in whom are equall angles • Here are fettwo definitions of like fedions of a circle. The one pertaineth to the angles whiche are fet in the centre of the circle and reeeaue the circumfercce of the fayd fedions: the other per¬ taineth to the angle in the fedion, whiche as be¬ fore was fayd is euer in the circumference • As if the angle B AC, beyng in the centre A and re- ceauedof the circumference B LC be equall to theangleFEG beyng alfointhe centre E and receaued of the cir cu inference F KG, then are the two fedions B CL and FGKIyfc® by the firft definition. By the fame definition alfo are the other two fedions like3naae* ly B C D,and F G H/or that the angle BAG is equall to the angle F E G. Alfo by thefecond definition if B A C beyng an angle placed in the Cir¬ cumference of the fedion B C A be e- angle E D F beyng an angle in the fe¬ dion E F D placed in the circumfe¬ rence, there are the two feCtionsBC A , and E F D'lyke the one to the o- ther . Likewife alfo if the angle BGC beyng in the fection B C G be equall to the angle E H F beyng in the fedio . E H F the two fedions BCG and E F H are lyke. And fo is it of angles beyng equall in any.poynt of the circumference. , \ v - - Bucllde* Euclide defmcth not equall Se&ions :for they may infinite wayes be defcribed. For there may vpponvnequall right lynes he fet equall Se&ions (butyet in vne- quall circles) For from any circle beyng the greater,may be cut of a portion equall to a portion of an other circle beyng the lelfe . But when the Sections are equall, and are fet vpon equall right lynes , theyr circumferences alfo flialbe equall * And right lynes beyng deuidedinto two equall partes , peroendicular Ivnes drawer* from the poyntes of the diuifion to the cir- cumfercces ihalbe equall. A s if the two fc£ti- oiis ABC and CD E F, beyng fet vppon equall ryghtlyn.es ACdcDF, be equall : then if eeh of the two lynes ACgcDF be deuidedinto twoe- quall partes in the poyntes G and //*,& from the uyd poyntes be drawento the circumferences two perpendicular lynes BG and EH, the fayd perpendicular lynes fhalbe equall ♦ ZfofThe ifProbteme . The i. Tropofition . To finde out the centre of a circle geuen. Vppofe that there lea circle geuen ABC . It is requi¬ red to finde out the centre of the circle ABC. (Draw in it aright line at all aduenturesy and let the fame he A B. And (by the i o* of the firf) deuide the line A B into two equall partes in the poynt 1 ). And(by the i i.of the fame ) | fro the poynt D raife yp lento AS a perpendicular line _ _ | D by the fecond petition )extend D C lento y point E. And( by the io*of the firft ) deuide the line C E into two equall partes ' the poynt F . T hen 1 fay that the point E is the centre of the circle ABC. For if it be not fet fome other po Gjbe the centre. And (by th tion)draw thefe right lines and GB. And forafmuch equall lento D By and ID G is lento the both, therefore thefe two lines A D and DG are equall to thefe two lines G D and D B?the one to the other find ( hy the . definition of the firf) the bafe G jC is equall to the bafe G B . For they are both drawen from the ceri* tre G to the circumference : therefore ( hy the 8. of the frft ) the angle ADO is equall to the angle BDG . But ~%>hen a right line Jlandmg lepon a right line maketh the angles on eche fide equall the one to the other y eyther of thofe angles ( by the io* definition of the firjl) is a right angle. WJeerefore the angleBDG yfa.iij. is 4 Why SaetlM defineih not equalised cm* CouflruZlkn* tion leading to an impofL fibilitk* €meUiy* lOrnon/lra* uo leading to an intpoffibi- Stie. The thirdflooke is a right angle : butj angle FID © is alfo a right angle by confiruBion. Vtfher* fore (by the 4 . petition)the angle FT) 'Bis e quail to the angle © T) Gythe grea¬ ter to the lejfe , K>hich is impofhble . Wherefore the poynt G is not the centre of the circle ABC, In like K>ife may ~fre prone that no other poynt befides F is the centre of the circle AFC. Wherefore the poynt F is the centre of the circle AT C: lohkh leas required to be done . Cor r el ary* Hereby it is manifefljhat if in a circle a right line do deuide a right line into two e quail partes, and make right angles oneche fide: in that right line which deuideth the other line into two e* quail partes is the centre of the circle* *r&The 1, T heoreme . The zfPropofition. If in the circuference of a circle be taff two poyntes at all ad- uentures : a right line drawen from the one poynt to the other Jhall fall within the circle. Vppofe that there be a circle A © C. And in the circumference tier* of let there be take at all adnentures thefe two poyntes A & ©. Then I fay that aright line dr awen from AtoT Jhall fall loithin the circle 'A © C.For if it do not Jet it fall without the circle, as the line AFT dotbptohich if it be pofiible imagine to be a right line . And( by the Tropojition going before)take the centre of the circle, and let the fame be D.Andfby the firfi petitwn)draw lines from ID to A, and from ID to ©„ And extend D F to E* And for afmuch as (by the 15* definition of y firfi ) T) A is equall lonto DT. T here fore the an* gle D A Eis equall to the angle ID © E.And for afmuch as o?ie of the fides of the triangle (DAE, namely the fide A ET is produced, therefore ( by the 16 ♦ of the firfi ) the angle (D E T, is greater then the angle D A EJBut the angle DA E is equall ynto the angle DTE. Wherfore the angle DET is great ter then the angle DTE. Tut (by the iS. of the firfi) Tmto the greater angle is fubt ended the greater fide . Wherefore the fide DTis -greater then the fide D K Tut(by the 15 ♦ definition of the firfi )the line D Tjs equall 0. FoLty, equaU lento the line 0F. VVhcrfore the line • D F is greater then the lint 0 Ey namely y the lejfe greater then the greater Ytohtch is irnpofiibk.Whtrforea right line drawenfrom AtoB falleth not without the circle. In like fort alfo may prone that it falleth not in the circumference : Wherefore it falleth within the , circle . If there fore in the circumference of a circle he taken two poyntes at dll ad* uentures: a right line drawenfrom the onepoynt to the other frail fall 1 whin the circle: U? hid: t>as required to he proued . y&The zfiTheoreme. The ifPropofmon. If in a circle a right line pafiing by the centre do deuide ano¬ ther right line not pa fling by the cetre into two equall partes: it Jh all deuide it by right angles . And if it deuide the line by right angles fit fh all alfo deuide the fame line into two e quail partes . Vpjmfe that there he a circle A B Cy and let there he in it drawen ybe&flHrt a right line pafiing by thexentye, and let the fame he C0> deuiding of this Aopo ah other tight line A B not pafiing by y centre into two e quail partes in the poynt F. T hen I fay that the angles at the poynt of the deuijion are right angles. T ahe ( hyy firfl of the third) the centre of the circle A B C\ and let the fame he E . And( by the firft petition ) drawe lines from E to Ayr from E to B> And for afmuch as the line A F is equall lento the line F'Byand the line F E is common to them bothy therfore thefe two lines EF and FA are equall lento thefe two lines E F& F B. And the hafe E A is equall lento the hafe E B (by the 15. defnk tion of thefirfi) . Wherefore( by the 8. of the firft) the angle AFE is equall to the angle BFE. But Sfrhen a right line Handing lepon a right line doth make the angles ■on etc he fide equall the one to the other yeyther of thofe angles is (by the 10. define tion of the firft)a right angle. VVherf ore either of thefe angles A FEyfy B F E is a right angle. VVherefore the line C 0 pafiing by the centre y and deuiding the line A B not pafiing by the centre into two equall partes ymaketh at the point of the deuijion right angles. But now fuppofe that the line C0 do deuide the line A B in fuch fort that it The feccnd ■maketh right angles. Then I fay that it deuideth it into two equall partes y that 'tsyythe line AFis equal! J>nto the line fiB. For the fame order of conftruction remay ?iing for afmuch as the line E A is equaU'tmto the line EB(hy the 15. de* til™” AaMij. finition Conjirutiiwo 'DtmonUra- tion. finkion of the fir ft), T herefore tlx angle EAF is e quail ^nto the, angle E ® F ( by the 5 . of the fir ft). And the right angle A EE is ( by the 4 * petition) equaBk to the right angle EE E. Wherefore there are two triangles E AF,Z? EEF hatting two angles equal l to two angles, £? one fide equall to one fide, namely the fide E F "Which is common to them botb,a?id fubtendeth one of the equall angles , "Wherefore (by the 26. of the firfi)the fides remayning of the one, are equdll y?t* to the fides remayning of the other . Wherefore the line jtF is equall Wn to the line F E . If therefore in a circle a right line pafiing by the centre do deu'ide an other right line not pafiing hy the centre intoXwo equall partes , it jhall deuide it by right angles . And if it deuide the line by right angles it Jhall alfo deuide the fame line into two equall partes : "whick"Was required to, be demonfirated. Theoreme. Ifihe ^fPropqfition. If in a circle two right lines not pafiing hy the centre > deuide the one the other : t hey jh all not deuide eche one the other into two equall partes* -Demn&ra- tion leading to an mpof- Jtbilitie* Fppofe that there be a circle AECD, and let there be in it drawen tw§ fright lines not pafiing by the centre and deuidingthe one the other ? and -- let the fame le A C and E ID, "Which let deuide the one the other in ths poynt E. T hen I fay that they deuide not eche the one the other into two equall partes , For if ithepofiible let them deuide eche the one the other into two equall partes, fo that let AE be equall lento E €,<&• E E Wnto E ID. And take the centre of the circle AECD , "Which let be F . And (by the firfi petition ) draw a line from F to E .Flow for afinuch as a certaine right line EE pafiing by the centre deuideth an other line A C not pafiing hy the centre into two equall partes, it maketh "where the deuifi* on is right angles (by the Z* of the third ) . VVherfore the angle EE A is a right angle. Againe for afinucb as the right line F E, pafiing by the centre j deuideth the right line E D not pafiingby the centre into two equall partes ,therefore(by the fame fit maketh "where y deuifion is right angles. Wherfore the angle EE E is aright angle. And it is proued that the angle FE A is a right angle. VVher- fore( by the 4 ♦ petition) the angle F E A is equall lento the angle F E E, namely the lejfe angle lento the greater : "Which is impofiible . Wherefore the right lines A C andED deuide not eche one the other into two equall partes. Iftherforem a circle two right lines not pafiing by the centre, deuide the one the other , they fhaft ofSttclides Elementes. Fol.%5. JhaUnot deuide eche one the other into two equall partes : 'which ~tyas required 2ft he demon f rated. In this Propofition are two cafes.For the lines cutting the one the other,do ey- yw0 Caj~es -tn ther,neyther of them paffe by the centre, or the one of them doth paffe by the cen- tins Propo- trc,& the other not.The firft is declared by the author.The fecond is thus proued, fttm. Suppofe that in the circle tA BCD the line B CD paffing by the centre doc cut the line tA C not palling by thecentre.Thenl fay that the lines C and B D do not dcuidc the one the other in¬ to two equall partes « For by the former Propor¬ tion the line B D paffing by the centre and deui- ding the line aA C into two equall partes, it lhall alfo deuide it perpendicularly.And for afmuch as the line A C deuideth the line B D into two equall partes & right angled wife:therfore by the Correb lary of the firft of thys booke,the line *A C palfeth by the centre of the circle : which is cotrarytothe fuppofitiom Wherfore the lines eA C and B D do not deuide the one the other into two equal! partes : which was required to beproued. ConflmBton for thepmid safe, • DemnUrfr Sion. 5-fc>T he 4.0 Theorem: The 5. Tropofitioru If two circles cut the one the other Jthey heme not one and the fame centre . \Vppofe that thefe two circles fBCj and CFG do cut the 1 one the other in the poyntes C and <B . Then I fay that they haue not one is* the fame centre. For if it be pofii* hie let E be centre to them both.Andfby the firf petition ) draw a line from E to C . And draw an other right line EFT at aldaduentures.And for afmuch as poyfit E is the centre of the circle AFC y therefore ( by the 1 5* definition of the firsfthe line E C is equall lento the line E F. yfgaynefor afmuch as the poynt E is the centre of the circle C T> G, then# forefby the fame definition)the line EC is equall Imto the line EG. And it is proued that the line E C is equall lento the line E F : therefore the line E F alfo is equall l?nto the line E Gqnamely the lejfe Imto the greater : lohich is impofil + ble. VVberfore the poynt E is not the centre of both the circles A IB Cgs* CFG . In like fort alfo may prone that no other poynt is the centre of both the fayd circles. CehjlrH&ktu Demon ft ra~ tio leading to mmpojftbi - litis a . Jm* 9. Dmwftra- Sion leading to an impQ]- in in thys Pro- fofithn. c inks. If therefore two circles cut the one the other 5 they haue not one mid the fame centre : ^bicb "Was required to be proued, S^Tbe j.Theoreme, The 6, *Propofitionm If mo circles touch the one the other , they haue not one and the fame centre, ' fppofi that thefe two circles A E Cyj? OD E do touch the one the other \in the poynt C. Then I fay that they haue not one and the fame centre . For if it be pofiible let the point F be centre Wnto them both. And (by the firfi petition)drdw aline from F to Ciand ( drawe the line FEE at all aduentures . j. fhdfor afmuch as the poynt F is the ten* tre of the circle A E C pther for e(by the I definition of the firH)the line F C is equal! Ipnto the line FE. (Agayne for afmuch as the poynt F is the centre of y circle C IDE f therefore (by the fame definition) the line FC is equally nto the line FE . And it is proved jhat the line F C is e quail Imto the line F E, "Wherefore the line FE alfo is e * quail lento the line F E} namely the lejfa yntoy greater:~which is impofiible. When fore the poynt F is not the centre of both the circles AEC and Cfb E . In like fort alfo may ive prone that no other poynt is the centre of both the fay d circles. If therefore two circles touch the one the other: they haue not one and the fame centre: "Which "Was required to be demonft rated. In thys Propofition are two cafes : for the circles touchyng the one the other, may touch eyther within or without » If they touch the one the other within, then is it by the former demonftrationmanifeft, that they haue not both one and the felfe fame centre. It is alfo manifeftif A they touch the one the other without : for that euery cen~ tre is in th e middeft of hys circle. ; . , . 1 j . .• • r .... i-^T'he6. Theoremc* The yfPropofition. If in the diameter of a ci many poynt. is not the ofSucIides Blementes* * FoLSd* the centre of the circle, andfromthat foynt he drawen ynto the circumference certaine right lines : the greate&of thefe lines (hall he that line wherein- is the centre , and tm wff fhad he the refidue of the fame line •And of alltbe other form, that whkh is nigher to the line which pajjeth by theeqntrejis greater then that which is more distant. Andfromthat point can fall within the circle o^eehfdefthekMi lineoneljtwo Vppoflybat time be a circle JfB- C®-:. mdJe&tfeidiameter thereof ke JJD . 'And take in it awyjpoynt hejides$et \ centre, of the. circle, ymd let the fame he K And let thexentre of the, circfifi by tj?e> i. * off third ) he the poynt & jfndfirom the poynt E let ; theyg he. draw w lento the circumference yfF C L D thefe right the line Fyfis thegreatefl : and the FID is the left . And of the other lines, the line F F is greater then the line F and the line F C is greater then the line F G . (Drawe (by the firfl petitionfhefe. right lines F Efi E} and G E. jTnd fo afmuch as( by the 2o» of the firfl fin ry triangle two fides are greater then the third therefore) lines E $ and E F are greater then the refidue , namely then, the line F F. Fut the line AE is e* qihlllmto the line FE(bythe i$*defi* nitionof 'the firftfiWherefore the lines BE and E Fare -equal! lento the line *AF. Wherefore the line ^AF is greater then then the line FF, Agayne for afmuch as the line F E is equal! pntoC E (by the 15. definition of thefirft)and the line FE is common Imto them both ^therefore thefe two lines FEand EF are equalhmto thefe two C E and EF. Fut the angle FEF is greater then the angle CEF. Wherefore ( by the 2 4..ofthe firfl) the bafe F F is greater then the bafe CF : and by the fame reafon the line CF is greater then the line FG. Agayne for afmuch as the lines G F and F E are greater then the line EG (by the zo.of the firfl ) . Fut( hy the 15. definition of the firftfihe line E G is equal! Ipnto the line E D : Wherefore the lines G F and F E are greater then the line E Djtake away E Fytybicb is comon to the bothptoherforey refidue G F isgrea * ter then the refidue FD ; Wherefore the line F A is the greatest }and the line FD is the lefl3and the line FF is greater then the line FC, and the lineFC ConflmBim* The firfl p*rt of this Props* fitlOHo Demonflrt* thfSo Second part. Third part . h greater then the line F G . Flow alfo I fay that from the poynt F there can he drawen onely two equall right lines into the circle jfBCD onechefide of the leaf line gamely F ID . For (by the 2% . of the firfl) lopon the right line geuen. B F and to the poynt in it, namely E, make "onto the angle G E F an equall an * gle FE H: and(by the firft petition)draw a line from F to H. Flow forafmuch as (by the 15. definition of the firft ) the line E G is equall Tmto the line E H, and: the line EF is common Wnto them both, therefore thefe two lines GE and EF areequalhmto the fe two lines HE and E F, and (by conftruclion) the angle G E F is equall into the angle HEF . Wherefore(by the 4. of y firft} This demon- thebafe FG is equall lonto the baj'e F H. I fay moreouer that from the poynt firmed by an jp Qm ye qrawen mt0 a'rc/e no other right line equall lento the line F G „ For dmg to an im- if it pofiible let the line F KJbe equall lonto the UneFG . And for afmuch as F If geftbiiie. }s equall lonto F G . (But the line F H is equall Imto the line F G, therefore the line F If is equall lonto the line F H. Wherfore the line which is nigherto the line which pajfeth by the centre is equall to that which is farther of which He haw before proued to he impofiible . iAn other de- Or els it may thus be demonftrated. ofT&tter bDraw ( by the firft petition ) a line from part of the E to If: and for afmuch as(byy 15* de* leadings find™ °fj firft)y He GE is equall lonto to an mpofii- J line & K.> dn^ doe fine F Eis common frilitit. to them botb}and the bafe G F is equaU lonto the bafe Elf, therefore (by the S. of the firft) the angle GEE is equaU to the angle IfEF. But the angle GEF is equall to the angle HEF . Where* fore (by the firft common fentence) the angle HEF is equall to the angle IfE F the lejfe lonto the greater : which h impofiible . Wherefore from the poynt F there can he drawen into the circle m other right line equall lonto the line G F. Wherefore hut one onely . If therefore in the diameter of a circle be taken any poynt, which is not the centre of the cir * demand from that poynt he drawen Imto the circumference certaine right lines : the greateft of thofe right lines Jhall be that wherein is the centre : and the leaft jhallbetherefidue . jindof all the other lines , that which is nigherto the line which pajfeth by the centre is greater then that which is more dift ant. And from that poynt can fall within the circle on ech fide of the leaft line onely two equaU right lines : which Was required to be proued. \ *4. CoroHary* fj H Corollary, Hereby ir is manifeft, that two right lines being drawen fro any one poynt of the diameteryJhe one of one fide,and the other of the other fide,if with the diame¬ ter they make equall anglcs,thc fayd two right lines are equal!. As in thys place are the two lines F G and FH. fTh ofSuclides Elementes . FoL 3 Jo y&T'he 7 . FTheoremc. 7 he S.Tropoftion . Ifwithouta circle be takpn any poynt, and from thatpOynt be drawn into the circle vnto the circumference certayne right lines ,of which let one be drawn by the centre and let the re fl be drawn at M adventures : thegreatejl ofthofe lines which fall in the concauitie or hollownes of the circumference of the circled s that which paffeth by the centre : and of all the other lines that line which is nigher to the line which pajfeth by the centre is greater then that which is more diflantfBut ofthofe right lines which end in the conuexe part of the circumfe¬ rence ,th at is the leaf which is drawen from thepoynt to the diameter: and of the other lines that which is nigher to the leafisalwaies leffe then thatwhich is more dif ant. And from th at poynt can be drawen vnto the circumference on ech fide of the leaf onely two equall right lines . 9 FfMd Fppofe y t&e circle geuen be A’B Cx without? circle AS C, take the ■ ^d^point ID : and froy fame point draw certain right lines intoy circle ynto the cir* cumference , zs let the be D A ,D E,D F, <Zsr D C:zsr lety line D A paffe byy centre. T hen I fay, ofy right lines H vhich fall in the concauitie ofy circumference A E FCy is, loithiny circle y greatef isy Svhich paffeth by y centre, that is, D [A. And of thofe lines vhich fall ypony conuex part ofy circumfe* rencefj) lef is y iohich is drawen froy point D lontoy end ofy diameter yl G. And of the right lines fallmglbin the circumferece , the line D E is greater theny line D E,Zy the line D F is greater theny line D C. jind of the right lines t>bicb end iny conuex part of the circumference J is, -without} circle ,that ybich is nigher yntoD Gy lef, is alwayes leffe theny 1* hich is more dipt, that is, the line D J\ is leffe then the line D E,and the line D E is leffe then the line D H. T ake ( by thefirfi of the third)thc centre of the circle J.B C, and let the Bb.j. farm The first part of this Propo - fimu. Sfcsnd part. Third part. fame be M : and ( by the firji petition) drawe thefe right lines' ME,MT?.JMC, MB, ML ?and M If. find for afmuch as (by the 15 .definition of the ftrft) the line MM is equall Icyito the line E M, put the line MD common to them both . Wherefore the line AID is eqnall Imto the lines EM and MD . ‘But the lines E M and M D are ( by the 2 o. of the ftrft) greater then the line E D: Wherefore the line MDaljo is greater then the line E D . Agaynefor afmuch as (by the definition of the ftrft) the line ME is eqnall lento the line ME, put the line MD common to them both '.Wherefore the lines EM andMD are eqnall to the lines FMandM D,and the angle E MD is greater then the angle/PM D : Wherefore (by the 19* of the firji) the bafie E D is greater then the bafie FD .In like jort alfio may Ice prone that the line ED is greater then the line C D . Wherefore the line D A is thegreateft,and the line D E is grea¬ ter then the line D Fj and the line D F isg, jfnd for afmuch as ( hy the 20. of the firji) the lines MJf and lfjD are greater then the line M D . ‘But (by the i^.defini* tion of the firji) the line M G is equal! J>n* to the line M If .Wherefore the ref due KJD is greater theny refidue G D.VVher* fore the line G D is lefife then the line KJD. .Andjor afmuch as from the endes of one of the fides of the triangle MED , namely, M D are drawen two right lines M If and KJD meeting within the triangle ,ther fore (hy the 21. of the firji) the lines M If and KJD are lefife then the lines ML if ED, of mhich the line M If is equal! Imto the line M E . Wherefore the refidue D If is lefife then therejidue D E . In like jort alfio may me proue that the line D E is lefife then the line D El . Wherefore the line D G is the left, and the line D If is lefife then the line D L, and the line D E is lefife then the line DH. Now alfio I Jay that from the pqyntlp can be drawen Tnto the circumference 6n eche fide of DG the leaft onely two equal! right lines . Vpon the right line MD, andlmto the poynt in it M make (by the 2%: of the fir ft) lento the an* gle KJMD an equal! angle DMfi . And (by the ftrft petition) drawe a line from D toB. Jbnd for afmuch as ( by the l^definition ofthefirU ) the lint M B is equall lento the line M- If put the line MD common to the both,mher* foye thefe two lines M If and M D are equall to thej'e two lines B M andMD the one to the other, and the angle LfM D is ( hy the 2\, of the ftrft) equal! to the angle B MD: IWherefore ( by the 4.*ofthefirU )the baft D If is equall FolM. . to the hafe D 3 . ISLow I fay that from the poynt D on that fide that the line ID 3 is, can not he drawen Imto the circumference any other line hefides D 3 equall lento the right line DAfePor fit he pofitble let there he drawen an other line hefides D tB, and let the fame be D N. y(nd for afinuch as the line D If is equall Imto the line D N. But lento the line D Jf is equall the line D3.Therfore ( hy the fir ft common fentence)the line D 3 is equall lento the line DK. Wherefore that ‘ft Inch is Higher lento ID G the least, is equall toy "Which is moredifiant: Which -We haue before proued to he impofiihle. Or it may thus he demonstrated » Draw (hy the firfl petition ) a line from M to 'N.- And for afinuch as (hy the 15* definition of thefirfi)the line IfiM is equall lento the line MN,and the line M D is common to them both . And the hafe JfD is c* quail to the hafe D IS! (hyfuppofition ) therefore(hy the ft * ofthefirfi ) the an* gle KfKFD is equal! to the angle DMpZ ,3ut the angle KfiAD is equall to the angle 3MD. JVberfore the angle 3 M D is equallto the angle N M W, the lejje lento the greater: -which is impofiihle : Wherefore from the poynt D can not he drawen lento the circumference A3 C on echefide of DO the left, more then two equall right lines . If there fore without a circle be taken any poynt and from that poynt he drawen into the circle lento the circumference certaine right lines, of -which' let one be drawen by the centre , and let the resihe drawen at all aduenturc-s : the greatefi of thofe right lines -which fall my concauitie or hollow * nes of the circumference of the circle is thatyhichpafiethby the centre . Jhdof all the Other lines , that line -which is nigber to the line -which pafieth hy the cen ! ire, is greater then that -which is more dittant.3ut of thofe light lines ' which end in the conuexe part of the circumference , that !me is the Mt -which is drawen from the poynt to the dimetient : and of the other lines that -which is nighef'tb the leaf is dlwayes leffe then thatyhicl: is: moredifiant . ^And from that poynt Can be drawen Imto the circumference on echfidtof thelefi only two equall right lines : -which -Was required to hefrbuedl y : r A"., ’ \ o*. KOiVa'.u ; A/ 'A \ck.v; . \ Ax~’1 ' v; Thys Proportion is called commonly in oldbookes amongeft the barbarous, C/udaPaupms,, that is, tfre Peacocks taile, \j 1 .y i,. - . i, \ J - > 'i v i i * A v ■ *\ ■‘•a \ • iJf ■ ; -■ / ' ’ ' * IS vv. wjtmm. i ■ " - TA yyyr - <y , t ■ -y , Hereby it is mani£efl>that the right lineSjWhich being drawen from the poynt Bb.ij. geuen This is demo* prated hy an argument (en¬ ding to an &b-' ’ fnrdity. An other de- monUratton of the latter part Reading alfo to an im- pofsibiiity» This Propon¬ ents cemmely called CatuU i*ttm>nssa Confiructioti. TXerwnftrtr t(on. K' gcuen without the circle, and fall within the circlc,are equally aidant from the lead, or from the greateft (which is drawen by die cehtre) are equal! the one to the other : but contrarywyfe if they be vncqualiy diftant,whcthcr they lightvpon rhe concauc or conuexe circumference of the circlc,they are vnequalh s y&The ZfTheorme. clhe yfPropofitm. f If within a circle he talpen a poynt , and from that poynt he drawen lemto the circumference rrioe then two e quail right linesithepoynttal^en is the centre of the circle. I Eppofe t hat the circle be A IB C, and loithin It let there be taken the poynt ID. And from ID let there be draieen lento the circumference ABC moe then two equal! rift lines , that is, D A,D B, andD C T hen I fay that the poynt D is the centre of the circle ABC \ Dram ( by the firfl petition ) thefe right lines JB and BC : and ( by the io» of the frfl fdeuide the into two equall partes m the poyntes E and F: namely, the line 4 B in the poynt E, and the line B.C in the poynt F , And drqpy lines y E D apd F Dy and (by the fccond pe* tition)extendthe lines E D and FD (}U eche Jide to the poyntes Jf, G,and fif . A nd for afmnch as the line AE as. equal! lento, the line E B,and the line E D is common to them both, , there * fore thefe two fdes A E and %D are equal! tynto thefe two J B D : and.( byfuppoftion ) the bafe D jf is equal! to the bafe D B . Wherfon (by the 8 . of thefirft) the angle A ED is equal! to the angle B ED Wherfore eyther of thefe anglepji E D aiid, B E D is 4 fight angle. Wherefore theftiig G Jfjleuidethy line A B into two equall partes andmaketh right angles. And for .afmuch as, if in 'a circle aright lipe denude an other right line into two equaJl parte? in fuel fort Eqtitniafjyqfrftafes^f y line that dcuideth isth cenjpe ft he circlfby the Coirolfy of ihefirfi oft he third) , T her fore ( by the fame Corr diary) _m the lino G fame reafon)maysxe prone that in y line HE is the. Centre of the circle A BC, and the right lines G Jf, and FI E bane no other poynt common to them (oth hejides the poynt D: Whereforedhe poyfit D is the centre of the circle A BC. Jf therefore within a circle be taken a poynt , and from that point bedfdweii th e circumference more then two equal) right hjties, the poynt taken is the centre of the circle: Svhicblvas reauiredto.be proued. ' ■ - ' •• tv •-•••• 4 ■- * - , *‘J ‘ ' /’ *• * " **J®- ' fjn ‘httni -x-atrk 1 7/ ■ c A a. . 3V.\ 0. . vh E:'. S I ides .rt.oa to an impofii bilitk* * FoL 8p. ft An other demonstration. ; ■ S ■ G\ ifcjk fdG 1 . k 0 k k -••• , ,v: r--:dkpff\ y‘ 'E'ei.thexehe taken ■^itUn-^^k(^:A.^€^he fojnt D.Anifrom. the poynt jn otforje, fDlet-thwe^Mdraw'eniyi^rf^&rcmfinxtckfndre then two equally tgbt Cmesy mnlimion namely } DA,DB} and 2) t.Then I fay thattbe poynt ID. is the centre of the: circle . For if not , then if it be pofible let the point E be the centre rand draw a line from Dio Eyand extend D E to the poyntes F and G . Wherefore the line F G is the diameter of the circle A EC. And for afmuch as in F G the diameter of the circle AFC is taken a poynt ? namely Dy 1 vhicb is not the centre of that circle , therefore ( hy the 7 . of the third) the lintDG is ygrea ohsVA TWAO teffy and the line D C'is greater then (y, . tea then the line D A. ‘But the lines D CfD Bfb A^are alfo e quail ( byfuppofi* tion) : which is impofihle . Wherefore the poynt E is not the centre of the circle ABC. And in likefortmhyWe prone that mother poynt befides D. Wherefore the poyrtt D is the centre of the circle ABC: “Which Was required to be troued, -x: ftjfcca jttivvl 5m»s tHima J 1 \ \pfipc wU'WA . WThe9M YUOrtu . . . dn <&m 'WV* e io. n , ■ •" V:> ::rav. option . s»' ■ i V‘ V;., E'E ■ k ■s\ RTOV' Id . j k A cm le cutteth not a circle in moepointes then two* *Vj ) tv i'.V '-.c •> V . ^ r . i «. f\ , ’ • > t i . . • 1 .’i ’ iff'1 y rt, ■ - \ Or if it le poflible let the circle A BC cut the circle D B F. in mopohites “""qLtben two float is pi BfGJ-dyW R, And drawe lines fro B to & ?and from tion lead in M B to rI.Ana(byy io .of the firffeuide either of the lines BG&B El into two equaU partes pi) pointes ' ....... j Jfand L. And by then* of the fir ft) from the poynt Ifraifelop yntoy line B II a perpendicular line KjC> and hkewife from the poynt E raife lop yntoy line B G aperpendi&dqr line hlf and . extend the line L jf to the poynt A y and EFIM to the poyntes IX and E .. And for qfitmch qs:in the circle' k/f BC 3 the right line ' A C deuideth the right line B IE rfei /srsv.- (hj r s A'V' ,-V 'L W ... v\, • \ \ ^ /x • • • l, . 1 o c jrEjfixC / 1 / . : ...,\ d'd maketh 'rght angles jherfore( by the 3 . of the third ) Bb.itj. in the An other de- ftion/l ration of the fume Heading alfo to an mpofti- kililie. m the line AC is the centre of the circle ABC. Agaynefor afmuch as in the jeife fame circle A BC the right line NXy that is Me line ME deuideth the right line B G into two equall paries andmaketh right angles ytherefore( Wjfthe third of the third) in the line NX is the centre of the circle ABC. And it k proued that it is alfo in the line , X C , 'And thefe two right lines A C and N X meete together in no other poynt befides 0. Where* fore the pqynt 0 is the centre of the circle ABC . And in like fort may 1 tie prone that the poynt 0 is the centre of the circle S)EF: Wherefore the two circles ABC and (D EF deuiding the one the other haue one and the fame cen * tre : yphiclf hy the 5 • of the third) is impofible . A circle therfore cutteth not a circle in moepoyntes then two^hich t>as required to be proued. ' ' ' ' 1 ■ ' ' m if An other detnonftration to proue the fame. S uppofe that the circle ABC do cut the circle (DGF in mopoyntes then that is jn Bf}f\and H, And ( hy the firfl of the third) take the centre of the circle ABC and let the fame be the poynt if. And draw thefe right lines IffB? JfGy and ]\F . Now for afmuch as within the circle !DE F is taken a cer* taine poynt and from that poynt are d'rawen ynto the circumference moe then two equall right lines y namely , KJBy i\G yand JfF : therefore (by the 9 ♦ of the third) If is the centre of the circle ADEF . And, the poynt If is the centre of the cirble ABC. wherefore two cir* des cutting the one the other haue one and the fame centre : "tyhich( by the £> ♦ of the third) is impofiible.X circle therfore cutteth not a circle in moe pointer then two : ‘tohich itas required to hi demom ft rated. 5 &T he IQ. Theoreme . The if fProftofition* If two circles touch the one the other inwardly , their centres being r-x ofEuclides Elementes. Fo/.po* inggeuen ; a right line ioyning together their centres and 1 will fall upon the touch of the circles. Vppofe thM jf&C, and ASJ-E do touch the one the other in the poynt A. And (by the firjl of the third) take the centre of ^ the circle A £> C^andlet the fame he F: and likpivifey centre oft he circle 4fD Ejand let tf e fame be Or-. Then l fay that aright line dnxwen from F to G and.behgprodnccifAill fall yponthe poynt j[. For if not 3 then fit he pofiible let it fall as the line F Q ID Fd doth . And dmwthefe right lines A F}i? A G. ror afmnch as the lines A G and QF are (by.the 2:o, of thefrsfgrea* ter then the line F A? that is, then the line F H fake aipay the line GF yehich is common to them both. Wherefore the ref due A G is greater then the ref due G EL . Tut the dine T) G is equall t>nto the line G A (by the W* definition of the fir f), Wherefore the line G dp is greater theny line G H: the leffe then thegreaterilohich is impofiible. Wher * fore a right line drawen from the poynt ftp fie poynt G and produced fal\eth not befides the poynt J, loUch is y point wal rxsl„vhfor'e itftllethypon the touch , If therefore two emits. imch quire d to beproued. ' \\ • .•*: r • ; • q ’ • <v ; Vv;r; v.“i v g-. : \ '-j - ;} An other demonftration to prone the (lime. Tut now let it fall as GF C fallethfnd extendy line G F Cto the poynt H: and dmwe thefi right lines J G and A F. Arid for afinuch as the lines JG and O F are ( by the 20» of the fir fl) greater then the line A F. Tut the line A F is equall y>nto the line CF, that is, ynto the line FH. Take away the line FG common to them both . Wherforethe refidue A G is greater then y refidue GH that is fire line GJD is greater then the line G H:the leffe greater then y grea *. ter : lohich is impdfiiblel ' 1 ft Which thing may alfo be proued by the 7.Propofitibn of this booIce.For for afmuch as the line H C is the diameter of tnc circle A B C ,8c in it is taken a poynt which is not the centre, nimely,the poynt G, therefore the line G A is greater then the line G H by the fayd y.Propofmon . But the line G D is equall to the line G A (by the definition of a circle) .Wherefore the line G D is greater then the line G H,namely}the part greater then the whole'-: which is irtipoffible. — - ■ - - 0 Conjlruftion, Demonftra- tion leading jiUlitti. An other de* man ft ration of the fame leading alfo to an mpeffla Wti'fi J he fame 4* gaine demon* jhated by an Argument lea « ding to an ab* fur d it it is. XU o • . -.0 rr * r '' 7 ■ / - t ■ '/ ' V&J > -v': | If wo circles touch the one the other ouwardly^a righflme , ' . . ri'G.W .K Av(pO; A\ kW ' \» V« C . . ' ‘ . ^ ' .'y I yTi"’ uV ig|| VppofewlMjT^fe tw&Sh'M A*B C and ft:lDE do touch flkone the o* And (by the third of the third') take the Umonjlnh tie leading to or/ tmpojfibi- titic . the right line F C ID G doth. And draw thefe right lines A Fcr AG. „ And for cijmuchas the poynt F is the centre of y circle JFB Cyber* free the line FA is equall Tmto the line FC . Again e for afmuch as the poynt G is the centre of the/ circle A iD E. , therefore the line G A is equall to the line G ID. And And it is proued that the line F A is equall to the Me VC . Wherefore the Tines FAmdAG dr/equall lento 'the -lines' F Cand'G"T>f Wherefore the Ipbole line poynt G jhall paffe by the poynt of the touch yiamefy }hy the poynt A . If there «*' fore two circles touch the one the other outwardly , a right line drawen by their centres Jhall paffe by the touch : "ft? Inch leas required to be demon f rated, • w ^ t * . • - ,L . _ , ff An other demorf ration after Telitarius. An other de - monfiration after Feliu- pj its leading at Jo to an ah- jurditie , V-\ \ \ iy Ka r-y Suppofe that the two circles ^4 B C and T>EF do touch the one the other out* watdly in the poynt fsfi And let G be the centre of the circle tABC : From wbic-h p.oynr produce by the touch of the circles the line G’^tothc poynt F of the circum¬ ference DEE . Which for afmuch as ir.paftetjinot by the centre of the circle 2> Efr (as the aduerfary affirmeth ) draw , from the fame centre G an other , right line G A', which - if itbepoffi* ) ble let paffe by the centre of the cir¬ cle D E F , namely, by the poynt Hi Cutting the circumference tABC F in the poynt B , & the circuference DEE in the poynt D y&let theop- pofite poynt therof be in the 'point X. And for afmuch as fro the poynt Cj taken without the circle D E F is drawen the line G K paffing by the centre M and fro the fame poynt is’drawen alfo an ©thee. / i ' AaT 0, mentes , FoL9 T, other line not patting by the centre,namely, the line G F . Therefore (by the § . of thvs bookc) the outward part G'T> of the lipe G K (hall bejeffethen the outward part G A oftheline G F.But the line G Ais equal! to the line G B .Wherfore the line G D islefie then the line O' ^namely ,the whole lefll* then the part ; which is abfurde. y&The 12. T bcoreme. The 13. Tropofition* A circle can not touch another circle in mce foyntes then one whether they touch within or without. Or if it be pofsible Jet the circle ABC D touchy circle EB FD firfi inwardly in moc poyntes then one ghat is fin D an dB. Trike I (by the firft of the third )the centre of the circle A(B C Dyand let 1 the fame bey point G;.and hkewjfej centre of the circle E'BFtD, and let the fame hey poynt H. Wherefore( by the 11. of the fame ) a right lh\e drawen from {he poynt G to the* poynt IT and, produced pttillf ill yp- on the poyntes B and D :■ let it jo fall as the line BG HD doth. And for ufmuch as the poynt G is the centre of the circle ABC Dgherejore (by the 15* definiti* on of the f rfi) the line B G is e quail to the line ID G , Wherfore the line B G is greater then then the line H D: Where fore the line B FI is much greater then, the line FID , Agrime for a f nuch as the poynt FI is the centre of the circle EBFD ^therefore (by the fame defini* trim) the line BH is equal! to the line HD : audit is proued that it is much greater then it: yphichis imjxfiible.. H circle thewfwO can not touch a cirelriin Of circlet Dehicb touch the one the other mp$ri~ & r a circle Pouchyth a circle iri-nioe poyntes then one. For if it he pofrible jet thy circle A C touchy circle A BCD outwardly in moc poyntes theme ,thafz- is jn A andC : And (by the firft petition)Arriw aline from the poynt \AtqtFT poynt C. Now for afmuch as in} circumference cfeith&(ftkf ci fries HBC ' and A C Ff^are taken two poyntes- at all adiknturesgAmefryJ^dTjhe^rr (by the fecofijfrtFe third) a righstfinetoynihg together tfoW p(yhtesjhal(fii- ^ithin both the circles . Bui itfalleth 1 mtlm the circle AB t.Dj? Without the circle AC KJ nvhichis abfurde ./ Wherefore a circle fhall not touch a circle, out* wardly in nioepointes then one Arid it is prouedy neither alfbinwardlyyyFereF yore a circle orimw&tp&Ghph other circle in moc: poyntes then oney Whether they t)f arch) H>hich touch the one the other out- fpkrdly. touch tvucbmthm or without: lohkh Tx>as required to be demonflrated. \An other de¬ monstration after Pehta- rtas zr Fluf - fates, of circles Jebicb toocb the one the ether out- ‘&4rdiy» Of circles which tooch the onr the other in¬ wardly. f ^Another demonstration after 4Pelitarius and Fluff ates. Suppofe that there be two circles AB G and AD G, which if it be poffible,iettouc& the one the other outwardly in moepoyntes then one, namely, in A and G. Let the centre of the circle AB G be the poynt I, and let the centreof the circle ADG be the poynt K. And draw a right line from the poynt I to the poynt K, which ( by the 12. of thys booke ) fhall pafle both by the poynt A and by the poynt G: which is not poffible : for then two right lines . Ihould include a fuperfiejes, contrary to. the laft common fentence. Itmayalfobc thus demonftrated.Draw a line from the centre- 1 to the centre K, which fhalipalfe by one of the touches, as for example by the poynt A. And draw thefe rigbtlines G K and G I, and/p (bail be made a tri- . '• <nglc, Vhofetwo fides G K and G I Hull not be contrary to the 2o.dfth,efirft. * Bi; brow if it be poffible, let the forefayd circle ADG touch the circle A B C inward- lyin’ woe. poyntesrthen bne,namelv,in the pointes A and G : and let the centre ©f the circle ABG be the poynt I , as before : and let the centre of the circle ADG be the poynt K, as alfo before. And extend a line from the poynt I to the poynt K, which lhall fall vpon the touch (by the 1 1 .of thys booke) . Draw alfo thefe line* KG, and IG . Andforafmucha^ theline KG is cqualltothelineKA(by the 1 5. definition of the firfl) adde the line KI common to them both. Wherefore the whole line AI isequalltothetwo lines KGandKI: but vnto the line A I is eq ual! theline I G (by the definition pf a circle)*Wber~,v fore in the triangle IKG the fide IG is not lefic ' then the two fides IK and KG; which is con¬ trary to the 20. of the firft. The i j. Theoreme. *The 14.. Tropofition. vimA \0 -Wtw'Alet The fir ft part of this Theo¬ rems. Conflruftion. In a circle, fe quail right lines ^eequ^diflatitfiim tre. And lines equally diTtantfrom tie centre, one to the other. w.iddv v' - ■ . . / \ cb k ' v > J a rffoj'e that there he a. circle { C there, he in it Tvy drawM : thefe equall right, fines 4 ® ® ^ hen lj^that-th^,4t’e[ equally dijiant, from, the centre, X <tke(hy ckf 4$ CdDyOfifUt thefd^h?yfoyntf[ ~ ^ . pint tk.line^4S<^eaX' ; , T ,t i e, c. - "V X'l • j ./ | s\ *.r W «-•* • ; .W 'wi -Vv perfmdicxkr ofEuclides Elemcntcs. Fol.p Am ft perpendicular tines EF and EG. And ( by thefuft petition) draw thefe right tines A E and C E . Flow for afmuch asa ccrtaine right tine E F drawen by the centre cuttetha certaine other right tine A B not drawen by the centre , in fuck fort that it make th right angles ^therefore (by the third of the third ) it deunleth it into two equall paries . Wherefore the line A F is equall to the line FB.Wher * fore the line A B is double to the line AF: and by the fame reajon alfo the tine C T> is double to the tine C G . But the line A B is equall to the line C D. Wher * fore, the line A F is alfo equall to the tine C G . And for of nuchas ( by. the 1 5 *de* fnitwn of the firf) the line A E is equall to the line EC, therefore thefquare of the line E C is equall to the fquare of the line A E . But fyhto the fquare of the tine A E,are equall (by the 47- of thefirft)the fquares of the lines A F W F E: for the angle at the poynt F is a right angle . And (by y felje famefto the fquare of the line E C are equall the fquares of the lines E G and G C:for the angle at the poynt G is a right angle . Wherefore the fquares of the lines AF andFE are equall to the fquares of the lines CG and G E: of lehicb the fquare of the line AE is equall to thefquare of the line C Gtfor the line A Fisrquall to the line C G . Wherefore ( by the third common fentence) thefquare remay ning , namely f the fquare of the tine F E, is equall to the fquare remayning , namely, to thefquare of the line EG : Wherefore the tine E F is equall to the line E G . But right lines are fay d to be equally difant fromy cen* trepyhen perpendicular lines drawen fro the centre to thofe lines, are equall (by the 4* definition of the third). Wherfore 'the lines A B and C D are equally difant from the centre. But nowfuppofe that the right lines AB and CT) be equally difant from the centre, that is, let the perpendicular line EF be equall to the perpendicular like EG. T hen 1 fay that the line A. B -is equal! to the line C T> . For the fame order of conftruRion remayningppe may in tike fort prone that the line A B is double to the line A F, dm that the line ChD is double to the line C G. And for afmuch as the line AE is equall to the line CE,for they are drawen fromy cen= tre to the circumference ,tber fore the fquare of the line. A E is equall tof fquare of the tine CE. But (by the 47. of tEffir§lfto thefquare of the line AE are equall the fquares of the lines E F and E A. And (by the felfe fame) toy fquare of the line C E are equall the fquares of the. lines E G and GC . Wherfore the fquares of the lines EF and FA are equall tothefquares of the lines EGaml G C. Ofylpch thefquare of the tine EG irequall to thefquare of the line E i\ for the tine E F is equall to the line E G, Wherefore (by the third common feu* tence) the fquare r ern ayn ing, namely, the fquare of the line. A E , is equall to the fquare of the line C G . Wherefore the equall '.into the tine C G. But ' ' the T : ‘0 ■ r\> JjcmonUra- tion. Demonstra¬ tion . Dhefecovd part which i she Conner je of the rip » ■An other de- monjiration c ftbefirft part after tampans ■GsnfirutUofi- the line AB is double to the line A F} and the line C 3) is double to the lint C G . Wherefore the line HB is equall to the line C 3) . Wherefore in a circle equall right l hies are equally diftantfrom the centre . And lines equally dijlant from the centre ^are equall the one toy other : h vhkh ~%>as required to be proued iff An other demonstration for the frf part after Cam fane. Suppofe that there be a circle nA B D C, whole centre let be the poynt £ . And draw in it two equall lines A B and CD. Then I fay that they arc equally diftant from the Centre. Draw from the centre vnto the lines AB and CD, thefe perpendicular lines E F and E G. And(by the 2 . part of the! 3 . of this booke Jthc line A B (hall be equally deuided in the poynt F, andtheline CD lhall be equally deuided in the poynt G. And draw thefe right lines EA,EBt EC, and ED. And for afmuch as in the triangle mAEB the two fides tA B and AE arc equall to the two fides C D and C E of the triangle C ED, & the bafe EB is equall to the bafe£Z>, therefore (by the 8. of the firft) the angle at the point A (hall be equall to the angle at the point C . And for afmuch as in the triangle A E F the two fides AE and A F are equall to the two fides CE andC<j of the triangle CEG, and the angl e.E A F is'equall to the angle CE G} therefore (by the 4. ofthe firft)thc bafe is equall to the bafe E G : which for afmuch as they arc perpendicular lines, therefore the lines ABSc CD are equally diftant fro thccentre,by the 4. definition of this booke, h&The Theoreme. The 15. Trofoftion, In a circle a the greatejl line is the diameter , and of all other lines tha t line which is nigber to the centre is alwajes greater \ then that line which is more diftant* Vppofe that there be a circle ABC 3), and let the diameter thereof be the line yf3>3and let the centre thereof be the poynt E. And ymto the diameter A3) let the Tine 3 C be nigher then the line F G . Then If ay that the line A3) is the greatejl ; and the line B C is greater then y line FG. 3) raw (by the 1 2 . of the frit) from the centre E to the lines B C and FG per* pendicular lines EH. and Elf . And for afmuch as the lineB C is nigher lan* to the centre then the line F Gptherfore of Bmlides Elemerites . Fol.py (by the 4 . definition of the third ) the line E If is greater then the line E H. find (by the third of the firft ) put lento the line E H ah equall line E L. And (by the u, of the firft) from the point L rdtfelep lento the line Ebfia per pen* dicularline LM: and extend the line L M to the poynt 7A . And(bg the first petition ) drata thefe right lines, E M, E N, E F, and E G . And for afinuch as the line E FI is equall to the line EE , therefore ( by the 14 *of the third, and by the 4 * definition of the fame ) the line © C is equall to the line M jSl. Againe for afinuch as the line A E is equall to the line E M, and the line .EE) to the line EM, therefore the line A A) ise* quail to the lines M E and E El . lint the lines M E andE !A are( by thezc, of the fir fi) greater then the line MIA. Wherefore the line yf A) is greater then the line MTA . Mad for afinuch as thefe two lines M E and E lA are equal l to thefe two lines F E and EG (by the 15, definition of the firft ) for they are dmwenfrom the centre to the circumfe¬ rence ,and the angle ME1A is greater then the angle F EG , therefore( by the 24. of the firft) the bafe MIA is greater then thebafe EG. Eut itisproued that the line MIA is equall to the line © C : Wherefore the line © Calfo isgrea* ter then the line F G . Wherefore the diameter ME) is the greateft ,and the line EC is greater then the line EG . Wherefore in a circle, the greateft line is the diameter ymd of dll the other lines, that line "which is nigher toy centre is alwaies greater then that line which is more diftant : which Was required to be proued. tfMn other demonfir ation after Campane . In the circle eABC I^whofe centre let be the poynt £, draw thefe lin es^BjAC, isf'D,FG3'j.nd HK, ofwhich let the line D be the diameter ofthecircie.Then I lay thattheline AD is .the greateft ofall the lines. And the other lines eche of the one is fo much greater then ech of the other ,how much nigher it is vnto the centre . loyne together the endes of all thefe lines with the centre , by drawing thefd right lines E B,E C,E g,E K,E £/,and E F. And ( by the 20. of the firft ) the two fides E F and £ G of the triangle EE G, (hall be greater then the third fide F G . Andforafmuch asthe fayd fides EF 8c EG are equall to the line AD (by the definition ofa circle) therefore the line tAD is greater then the line ££?. And by the fame reafon it. is greater then euery one of the reft of the lines, if they be put to be bafes of tri¬ angles ; for that euery two fides drawen fro the Oc.j. centre Demonjlnt* lion. An other de-* monftr ation after Cam - 1 fane* / centre .are equal! to the line t/C D . Which is the firft part ofr* the Propofition. Agayne,for af- irmch as the two (ides EF and EG of the tri¬ angle E F Gy are eqaall to the two fides E H and EK of the triangle EH K, and the angle F EG is greater then the angle H E X,therforc (by the 24*ofthe firft) thebafe FG is greater then the bafe H K. And by the fame reafon may it be proued, that the line A C is greater then the line A B . And fo is manifeft the whole Pro- $>ofitidn. " r't- V * * ‘ • w . <: . > •' s ... ... T. , ... , . . Si /The 15. Theorems. T'he \6.Tropofition . If from the end ofthe diameter of a circle be dr often aright line making right angles : it (hall fall ftithout the circle: and betftene that right line and the circumference can not be dr Often an other right line : and the angle ofthe femicvcle is grea ter then any acute angle made of right lines ,but the 0^ ther angle is lejje then any acute angle made of right lines. The foft part of thts Theo- reme* T>emonftra- tv'-n leading Appofe that there he a circle AB C: whofie centre let he the point D9 and let the diameter therofhe AB.T hen I fayy a right line drawen \ from the poynt A, making with the diameter A I B right angles fh all z fall without the circle . For if it do not /hen if it he pofiihle 9 let it fall Within the. circle as the line A C doth, ; and draw a line from the point D to the point C. „ And for afmuch as (by the ifT* definition of the firfl) the line DA is to an ahfurdi- equal! to the line D C ?for they are drawen from the centre to the circum * ference 3 therefore the angle DAC is equall to the angle A C ID. (But the an* gle ID AC is (by fuppofition)a right angle : Wherfore alfo the angle ACID is a right angle . Wherefore the angles D AC and AC D /re equall to two right angles : which (hy the 17. of the firft) is impofiihle . Wherefore a right line drawen from the poynt A > making With the diameter A B right angles frail not fall within y circle. In like fort alfo may We prone /hat itfalleth not ih ofSumctes Elenientes. FoL<)-\. the circumference . Wherefore itfalleth ydthout,as the line A E doth . I fay alfo,that hetwene the right line A E, and the circumference AC By can not he drawen anothcrrlght line-. For f it be pofiiblejet the line AF fo he drawen . And (by. the 1-2. of the fir ft) from the poynt ID draw tinto the line FA a perpendicular line D G . And for afinuch as AG D is a right angle , but DAG is leffe then a right angle , therefore (by the 19*0/' the firftfihe fide A D is greater then the fide D G . 'But the line D A is equall to the line D FI, for they are drawen from the centre to the circumference . Wherefore the line D H is greater then the line D G: namely, the leffe greater then the greater : ivhich, is impofiible . wherefore hetwene the right line AF. and the circumference ACB, can not be drawen an other right line. 1 fay moreouer , that the angle of the femicircle contayyied "tindery right line A B and the circuference C HA, is greater then any acute angle made of right lines . And the angle remay ning cotayned "tindery circumference CHA and the right line A E,is leffe then any acute angle made of right lines . For if there be any angle made of right lines greater then that angle Hnchiscon* tayned "tinder the -right line BA and the circumference C HA, or leffe then that K>])ich is contayned "under the cir * cumference CHA and the right line A ft, then bettime the circumference CHA and the right line A E, there /hall fall a right line , "tihich maketh the angle contayned Under the right lines, grea* ter then that angle "tihich is contayned "tinder the right line BA and the cir * cumference C II A, and leffe then the angle "tihich is contayned Under the cir * cumference, C H A and the right line A E. But there canfallno fuch line, as it hath before bene proued. Wherfore no acute angle contained tinder right lines , is greater then the angle contayned tinder the right line B A and the circumfe * rence C H A, nor alfolejfe then the angle contayned tinder the circumference. C H A and the line A E. Correlarj . Hereby it is manifefi that a perpendicular line drawen fro the end of the diameter of 4 circle . toucheth the circle: and that a right line toucheth 4 circle in 0 ne poynt onely . For it was proued (* hy the If of thethirdfihat a right line drawen from two pointes taken in Cc.ij. the Second path 7 hird parti CoftJirH&htt. Vmonftra- mn. An addition ofPtlttmns. the circumference of a circle, (hall fall within the circle. Which was required to be demonjlrated, 5 ^The 2. Trobleme. The 17. Tropoftion0 From a poyntgeuenjo draw a right line which Jhall touch a circlegeuen • j Tppofe that the poyntgeuen he jf,and let the circlegeuen he SOD. It is required from the poynt .A to draw a right line "Which Jhall touch the circle BC D . Take (by the firjl of the third) the centre of the circle }and let the fame be F. And(by thefrfl petitionjdraw the right line AIDE. And making the centre F, and the space ylF, defcrihe ( hyy third petition ) a circle .AF G. And from the poynt D raife 1 ?p( by the 1 1 *of the firjl ) nto the line E A a perpendicular line D F. And (by the firjl petition ) draw# thefe lines EftF and A ft . Then I Jay, that from the point A is drawen to the circle BCD a touch line A ft. For for of much as the point E is the centre of the circle BCD , and alfo of the circle A F Gyherfore the line F A is equall to the line E F, and the line F D toy line F ft, for they are drawen from the centre to the circumference. Wherefore to thefe two lines AF and E ft, are equall thefe two lines E Fir F D, and the angle at the poynt F is common to them both : Wherefore ( by the 4 . of the firjl) the baje DF is equall to the bafe A ft,andy triangle DEF But the angle ED F is a right angle : Wherfore alfo the angle Eft A is a right angle , and the line F ft is drawen from the centre . But a perpendicular line drawen from the end of the diameter of a circle, toucheth the circle (byy CoreUa» ry of the \6.of the third ) . Wherefore the line A ft toucheth the circle BCD. Wherfore from the point %euen, namely, A, is drawen Tfntoy circle gene ft C D9 a touch line yl ft : "Which "Was required to be done. jfy addition offtelitarius , Vnto a right lyne which cutteth a circle, to drawe a parallel line which lhaK touch the circle, * Suppofe es* Folpfe Suppofethat .the right lyije. AB do cut the circle AB Cinthepoyfites AandB* It is require^ todrawe vntpthelineAB a parallel lyne & F H which ihall touche the circle.' Let the centre of the circle be the point D. And deuide the lyne A B into two equall partes in the point E. And by the point E and by the centre D, draw the diameter CDEF, And from the point f'(whlch is the ende ofthe dia¬ meter) ray fe vpf by the 1 1 .of the fir if) vnto the dia¬ meter Cl F a perpendicular line G F H. Then I fay rthatthe lyne G F H (which by the correilary of c he 1 <?«.of this booke toucheth the circle) is a parallel vnto the line A B.For forafinuch as the right line C Ffallyng vpon either of thefe lines AB & GHma- keth all the angles at the pbint E' right angles (by the i ,of this bokejand the two angles at the point F are luppofed to be right angles : therforef by the 2 p,of the firft) the lines A B and G H are parallels : which was required to be done.And this Probleme is very cominodious for the inscribing or circumfcribing of figures ifl or about circles. cl he 1 6, Theorem^ This Pn- btems cimw-fc. Amu fa? %m in farthing c.-A ctrtumjen- Hngofpgur-n m or about sirties* onion* .v . ,, • f ffa right lyne touch a circle, and from the centre to the touch be dr amen aright line/ that right line fo dr amen Jhalbe a per* pendicular lyne to the touchelyne* jfippofe that the right line ID F do touch the circle ABC in the point ‘Dmonfln- And take the centre o f the circle ABC, and let the fame be F.And mn . fffgfh the jiff petition ) from the poynt F to the poynt C dr awe a right ^ dme FC . T hen I fay, that C Fisa perpendicular line to D EcFor if J s not ,draw(hy the i z.of the ftp) from the poynt F to the line D E a perpendicular line EG. .. And for a fmuch as the angle FGC is a right angle, therefore the angle GCF is an acute angle: Wherefore the angle FG C is greater theny angle FCG, hut Imto the greater angle is Jubtended the greater [ide( by the 19. ofthe firft). Wherefore the line FC is greater then the line F G. 'But the line F C is equall to the line F'Bjfor they are dr awen from the centre to the circumference: Wherfore the line FB alfo is greater then the lineFG, namely /he kffe then the greater.^hich is impojlihle. Wherefore the line EG is not a perpendicular line lento the line D F.And in like fort may K>e proue/hat no other line is a perpendicular line Imtoj line D F bejides the line F C: Where¬ fore the line FC is a. perpendicular line to DE. If therefore a right line touch jn - ' * ' ■' - Cc.iij. An other de- monflration after Qrsn~ tint. ^Iheihird^Booltp 4 circle ^frorfiy centre to} touch be drawena right line y right line fo dr amen fall be a perpendicular line toy touch line : lohich Teas required to be proued. f An other demonflration after Orontius . Suppofe that the circle geuen be A B C, which let the right lyne D ton ch in the point C» And let the centre of the circle be the point F.And draw a right line from F to C.Then I fay that the line FC is perpendicular vnto the line DE. For if the lineF C be notaperpediculer vnto the line D E, then, by the conuerfe ofthe x. defi nition of the firft boke,the angles D C F & F C E fbal be vnequalb&: therfore the one is grea¬ ter then a right ang!e,and the other is lelfe then a right angle , (Tor the angles DCF and F C E are by the x 3 .of the firft equall to two right an¬ gles) Let the angle FCE,ifit be poffible,be grea¬ ter then, a right angle, that is, let it be an obtufe angle.Wherfore the angle DCF fhal be an acute angle. And forafmuch as by fuppofitio the right lineD E toucheth the circle AB C , therefore it cutteth not the circle. Wherefore the circumfe¬ rence B C falleth betwene the right lines DC&CF:& therfore the acute and re&iline angle D C F (hall be greater then the angle ofthe femicircleBC F which is contayned vnder the circumferece B C 8c the right line C F.And fo fhall there be geue a re&iline & acute angle greater then the angle of a femicircle: which is contrary to the 1 6 , propo¬ rtion of this booke.Wherfore the angle D C F is not leffe then aright angle, In like fort alfo may we proue that it is not greater then a right angle.Wherfore it is a right angle, and therfore alfo the angle F C E is a right angle.Wherefore the right line F C is a per¬ pendicular vnto the right line D E by the 10. definition of the firft ; which was required to be proued. \ _• > - ^The \ jfTheoreme . fhe\<y. Tropoftion. ffa right lyne doo touche a circle, and from the point ofthe touch he rayfed vp vnto the touch lyne a perpendicular lyne, rath at lyne Jo rayfed vp is the centre ofthe circle . Demonftra - tion leading to an impo/0 fbilkie. rife that the right line D | touch the circle ABC in j C, And from C raife'Tpfyy n.oj the frjlfnto the line IDE a perpendicu* lar line C A. Then I fay jth at in the line 3 C A is the centre of the circle . For if not, then fit be pofible } lety centre be Trith* out the line C A }as inj poynt F And ( by the firjl petition) draw a right line from C to F. And for afmuch as a certaine right line D E toucheth the circle A © C?and from the centre to the touch is drawena right line C l\ therefore ( by the i8» ofthe third) F C is a perpendicular line to \ • ' • ' ' DE . ofSuclides Elementes * Fol,$6* 0 E. Wherefore the angle FCEis a right angle . Eut the angle jfCE isalfo aright angle: Wherefore the angle EC E is equal! to the angle A CE, namely s the lejfe Imto the greater : yohich isimpofible Wherefore the poynt F is not the centre of the circle A E C, And in like fort may ypeproue ,y it is no other Cohere hut in the line A C . If therefore a right line do touch a circle, and from the point of the touch he raifed Ipp Imto the touch line a perpendicular line , in that line fc rat fed lop is the centre of the circle : y&hich % vas required to he proued , f^pThe 18, clheoreme% TheiQ.fPropoJitwn. In a circle an angle fet at the centre Js double to an angle fet at the circumference ,fo that both the angles haue to their bafeoneand the fame circumference. Vpfofe that there be a circle ABC, and at the centre thereof , namely } the poynt EJety angle EEC he fet,& at the circumference let there be jet the angle E A C, and let them both haue one and the fame bafe, namely, the circumference EC, T hen I fay, that the angle EEC is double to the angle E AC, 0rawj right line A E,and(by the J'econd petition fx* tend it to the poynt F. Flow for afmuch as the line A E is e quail to the line E E, for they are drawen from the centre lento the circumference , the angle EAE is e* ^ quail to the angle EE A( by the S * of the firfl ) . Wherefore the angles EAE and E E yf are double to the angle EyfE, Eut ( by the Z1* of the fame ) the angle EEF is equall to the angles E jdE and E E yf: Wherefore the angle EEF is double to the angle E A E . And by the J'ame reajon the angle F EC is double to the angle E ylC. Wherefore the “tohole angle EEC is double to the ivbole angle E A C, KAgaine fuppofe that there he fet an other angle at the circumference , and let the Jame be EE) C. And(by the firft petition)draw a line from 0 to E. And (by the fecond petition)extend the line 0Elonto the poynt G . yfnd in like fort may ice proue,that the angle G EC is double to the angle E 0 C. Of yohich the an * gle G EE is double to the angle E0E. Wherfore the angle remayning E EC ts double to the angle remayning E0 C. Wherfore in a circle an angle Jet at the centre, is double to an angle fet at the circumference fa that both the angles haue to their hafe one and the fame circumference : lehich yeas required to be demon* sirated, ' Ccfnij. z’& 'Tbe T Wo cafes in thys Pro* f option the one when the angle jet at the circumfe - renceinclu- deth the cen~ ter* VemonfttA- mn . The ether wbe the fame angle fet at the circumfe - renccinclu- deth not the center* Irhff hc ip. 'Theorems. fths zi. TroPofitwn. ' vj~ " - ‘ •'■v • •'' ' ■ ■ - : l .< V-'V-i v%-'*yv.V' i! In a circle the angles Tvhich conjlsl in one and the felfe fame JeBlon or figment ,are equall the one to the other , ‘ \ ’ ' v- * ' 4 w •/-. Mr 1 V “ * * •;* a,**“->* : : o i \ . •••■’■ .C.<;C - A ? ■: ; ; Vppojey there be d circle A3 C Dps in the fegment therof'B A E D, let there con ftfi thefe tingles 3 AID and 3 ED . Then I fay, that the angles 3 AD and 3 ED are equall the oneto the other. Take(by the fir ft. of the third ) the centre of the circle Conjlmtm* yd 3 CD,and let the fame be the point F. Andfhy thefirfipeiition)drct\v theje lines pemon first* <J$p anp FT), ISLow for afinuoh as the angle 3FD is fet at the centre, and the angle 3 AD at the circumference , and they bane both one andy fame bafeyiame* lyphe circumference 3C D .therefore the angle 3 FD is (by the Tropofition going before ) double to the angle 3 AD : and by the fame reafon the angle 3FD is aU fo double to the angle 3 ED. Wherefore ( by the 7* common fentence ) the angle 3 jdD is equall to the angle 3 ED .Wherefore ina circle the angles “Which cohfifte in one and the felfe fame fegment yire equall the one to the other : 1 vhich "Was required to be proued. Three cares m this Propefi- tion. The firti cefe. The fecond uje. 1b this proportion are three cafes.For the angles confifting in one and the felf fame fegnienr,the fegment may either be greater the a femicircle, or lefle then a femicircle, orelsiullafemiciycle^Forthe firfl cafe the demonftration before put ferueth. But now fuppofe that the angles BzAD and B E Dd o confift in the fe&io BAT), which let be leffe then a femicircle. Euen in this cafe al¬ io I fay that the angles 2? A D and B ED are e- quail. For draw a right line from A to E. And let the lines A D and BE cutte the one the other in thepoynt(7, wherefore the fegment cACEis greater then a femicircle. And therfore by the firfl part of this propofition the angles whiche are in it, namely, the angles %A T E and EDA are equall the one to the other. And forafmuch as in the triangle eA B G the inward and oppo¬ site angles (7 and Cj A B are equall to the outwarde angle B G D, and by the fame reafon the two angles E D (7 and G E D of the triangle D EG are equall to the felfe fame outward angle B GD. Wherfore the two angles A B G and G A B are equall to thc-Wo angles ED G and G ED by the Fo/.p 7. by the firft commofentenec.From which if there be taken equall angles, namely, j4B G , and E <D G} the angle remainyngZf-^G’ (hall be equall to the angle remayning D E G, that is,theangle B A D to the angle D E B( by the third common fen- tence) which was required to be proued. The felre fameconftru&ion and dcmonftrati- ^ cn will alfo ferue,if the angles were fet in a femi- circle as it is playue to feejn the figure here fet. 0 The third s&fe* fr&The lo.Theorcme, The 21* Trope fit ion. If within a circle be deferibed a figure of former fides, the an¬ gles therof which are oppofite the one to the other > are equall to two right angles . f ^PP°fe ^?at there. a circle. ABC ID, and let there he deferibed in it i^nX^ a figure of fiver fides , namely , A B CD. T hen I fay ,tbat the angles thereof finch are oppofite the one to the other, are equall to two right ■ — | — 'angles. Draw (by the firft petition ) thefe right lines A C and B D. Conjhufih® Tdow for ajmuch as (by the 3 i* of the firft) the three angles of euery triangle are equal l to two right angles: therforey three angles of the triangle ABC, namely . Demon CAB,ABCyandBCA,areequaUto ***. two right angles . But (by the zi. of the third ) the angle CAB is equall to the angle B DC for they conffi in one and the felf fame fegmet, namely, B ADC. And (by the fame Tropofition) the an* gleACB is equall to the angle A DB, for they conffi in one arid the fame fig * merit ADC B . Wherefore the K>ho/e angle AD C is equall toy angles B AC and ACB: put the angle ABC com * monto them both. Wherefore the angles A B C,B A C, and A C B,are equall to the angles ABC and ADC. But the angles ABC, B AC, and A C B, are equall to two right angles . Wherefore the angles ABC and ADC are equall to two right angles . And in like fort alfo may ive prone, that the angles BAD and DCBare equall totworight angles: ’Demonfird- tion leading to an impof- fibilme . An addition cf Cam pane dsmonjlrated by Pdsiarms. "Demonjlra tivn leading to an impofii- bilitts , - ' ^ i f J(v - . 2/ therefore ‘toitbin a circle be defcrihed a figure offower fides, the an* gles thereof ~%>btcb are oppofite the one to y other, are equal! to two right angles: Tvhich t>as required to he proued. >-&'! he zuT heoreme , The 1 3 . Tropofition . cOpon one and the fife fame right line can not be defcrihed Wo like and vne quail fegmentes of circles falling both on one and the felfe fame fide of the line. Or if it he pofiihlejet there he defcrihed lapon the right line A B two like zjr Imequall fe Elions of circles , namely , ACB iy ABB, falling hoth on one and the felfe fame fide of the line J B . And (by the fir ft petition ) drawe the right line JC B, and( by the third petition) drawe right lines from C to B, and from B to B. .And for afinuch as the figment ACB is like to the figment jC B B : and like fegmetes of circles are they dnch haue equal l angles ( by the io, definition of the third). Wherefore the angle A CB is e quail to the angle ABB , namely , the outward angle off triangle CBB to the in ward angle : ~$>bicb( by the 1 6* of thefirft) is imqmfiihle.Wherfore lop* on one and the felf fame right line can not he defcrihed two like ty Unequal! peg* mentes of circles falling both on one ty. the felfe fame fide of the line: T vhich "tyas required to be demonjlrated. Here Campane addeth that vpon one and the felfe fame right lyne cannot be deferibed two like and vnequall lections neither on one and the felfe fame fide of the lyne,nor on the oppofite fide.That they can not be delcribed on one and the felfe fame fide3hath bene before demonftrated3and that neither alfo on the oppo¬ fite fide3Pelitarius thus demonftrateth. V A B . Let the fe&ion A B C be fet vppon the lyne 'A C,and vpon the other fide let be fee the fe&ion ADC vppon the felfe fame lyne A C, and let the fe&ion ADC belyke vnto the fe&i¬ on ABC. Then I fay that the fe&ions A B C and A D C being thus fet are not vnequal.Forifit be pof- fible let the fe&ion A D C be the greater. And de- uide the line A C into two equal partes in the point A E. And draw the right lyne BED deuiding the lyne A C right angled wife. And draw thefe right lynes A B, C B3A D and C D, And forafmuch as the lection A D C is greater then the fe&ion A B C, the perpen dicu.lar lyne alfo E D (hall be greater then the per¬ pendicular lyne E B : as is before declared in the ende of the definitions of this third booke* Wher- ofSuclides Elemmtem Foh p8 * fore from the ly ne E Dgut of a4y netiquaj.l tp the Iyne E B s which let be E F. And draw tfiefe right iynes A F and C F.Now' then(by the 4*of the firlljthe triangle A E B (hall be ecpiall to the triangie.A E'F,and the angle E B A iliall be equall to the angle E FA. And by the fame reafon the angle E B C (hall be equall to the angle E F G . Wherefore the whole angle A B C. is-equail to the whole angle A F C.But by the z i. ofthefirfigthe an¬ gle A F C is greater then the angle A D C.Wherfore alfo the angle A B C is greater then the angle A DC. Wherefore by the definition the fedtions AB Cand AD Care not iyke,which is contrary to the fuppofition. Wherefore they are not lyke and vnequall ; which was required to be proued. S^The iiLTheorme , TThe i^SPropoftion. Like fegmentes of circles defcriledvfipon equall right lines, are equall the one to the other . Wppofe thatypon thefe equall right lines AB and CD he deferibed P thefe like fegmentes of circles /Lamely > AF B and CFO). Then I fqy} ^ that the figment. ABE is equall to the ferment CFD . For putting the fegment A E B 'Opon the figment C FDynd the poynt A'Opon y pojnt C} and the right line A B hpon the right line C D } the poynt B alfo (hall fall Tvpon the poynt Dfiory line A B is equall to the line C D.And the right line AB ex> aFily agreing Soith the right line CDfi E F fegment alfo AEB Jhall exaFlly agree "S nth the fegment CFD . For if the right line AB do exactly agree loith /i the right line C D , andthe figment AEB do not exaFlly agree *t»ith the figment CFD? but dif¬ fer eth as the figment CGD doth: Now(by the i of the third) a circle cutteth not a circle in more pointes then two fit the circle CGD cutteth y circle C FD in more pointes the two y hat is yn the points Cfiynd D:K>hich is(by the fame ) impofiible .Wherefore the right line AB exaFlly agreing leith the right line C Dy the fegment AEB Jhall not hut exactly agree hath the fegment CFD: Wherefore it exaFlly agreeth K>ith and is equall 'Onto it . Wherefore like fig* mentes of circles defer ibed'Opon equall right lines ,are equall the one to the other: ‘ Hebich "Mas required to be proued. ThisFropofition may alfo bedemonftrated by the former propofition. For if the fe- &ions AEB and CFD being like and fet vpon equall right lines esf B and C‘Z>, ihould be vnequailythen the one beyng put vpon the other, the great; r iliall exceede the leife: but the line A A is one line with the line C'Di Cq that tlierby ihalfolloiy the contrary of the former Propofition. an A ' Belitarim Demonftra-* tion leading loan impoF fibiline. An other de-* monjiration. *An other /#- monjlrAtio $er Fslitmnu Conjlmtlkn, three cafes in skis Frapo/i- tiott. the prft cafe, V V , ' v, DymmUrA- Sion* Belitarius demonUrateth this Proportion an other 7$ ay. Suppofe that there be two right lines tsf B ScC D which let be equall .• and vpon the let there be fet thefelike fcftions A B /^and C D E.yhcn I fay that the faid fedions are equall.For if no when let C E D be the greater fedion. And deuide the two lines *A B and CDinto two equall partes,thcline A Bm the pointed, and the line CD in the point-G’.And cred two perpendicular lines F ifand G A.Anddraw thefe right lines A XdcKB; EC}ED. Andforafrauchasthe fedion CAD is the greater, therefore the pcrpendicularline GDIs greater then the perpendicular F if: From the lyncG E cut of a line equall to the line fifjwhich let be G H : and draw thefe right lines C i/and H DfAnd forafmuch as in the triangle *A IC F the two iides A F and F K are equall to the two fides CGScG H of the triangle CH~ G,and the angles at the pointes A and G are eq ual ( for that they are right angles) thcr- forefby the 4. of the firft)the bafe K is equall to the bale CH, and the angle A K F to the angle CHG.And by the famereafon the angle B K Fis equall to the angle D H G. Wherfore the whole angle A KB is equall to the whole angle C H D.But the angle CH Dis greater then the angle C ED by the 2 1 .of the firft*Wherfore alfo the angle A KD is greater then the angle C E D. Wherfore the fedions are not lyke, which is contrary to the fuppofition. •4 The 3 . Trobleme . The 25. Tropofition . ctJ figment of a circle beynggeuen to deferibe the whole cir > c/e of the fame fegment . Vppofe that y figment geuen be ABC.lt is required to dejerihe the yohole circle of the fame fig* ment\AB C. Deuide (by the 10. of the frjljtbe line A C into two.equall partes m the poynt D . And (by the 1 u of the fame) from the poynt D raife yp ynto the line A C a perpendicular line B D. And ( by the frit petition) draw a right fine from A to \BJSLow then the angle ABD being compared to y angle rB A (Dp is either greater then itfir equall lento ityr lejfe then it. First let it he greater, And ( by the 23 ♦ of the ftmeftpon the right line B As and ynto the poynt in it A, make ynto the angle ABD an equall angle B A E. And(by the fecond petition)extend the line B D ynto the poynt E. Andfby the fir ft petition) draw a line from E to C. Nora for afinuch as the angle ABE is e* quail to the angle BAE, therefore (by the 6 ♦ of the firfi)the right line EB is equall to the right line A E . And for afinuch as the line AD is equall toy line D C and the line D E is common to them both : therefore thefe two lines A 2) and of Suclides Elemerites * FoLpp, and DE} are equall to thefe two lines C ID and D E the one to the other . And the angle .A D E is ( by the 4 * petition') equall to the angle C D Ey for either of them is a right angle . JV here fore ( by the 4 . of the firft ) thebafe AE is equall to the bafe C E . 'But it is proved ghat the line A E is equall toy line B E.Wber* fore the line BE a jo is equall to 'the lineC E. Wherefore thefethree lines A Ey E Byand E C yare equall the one to the other . Wherefore making the centre E? and the space either A Ey or EByor E C: defcribe (by the third petition) a cir* demand it fallpqffe by the poyntes Ay By C . Wherefore there is defcribed the ivhole circle of the fegment gemn.. And it is mamfeffthat the fegment ABC is lejfe then a femicircle for the centre Efalleth without it* Tlx like demon f , ration dlfo Will feme if the angle ABD be equall to the angle B AD . For the line A D being equall to either of the fe lines y B Dy and D Cy there are t hree lines y D A yD By.ndD Cy equall the one to the other . So that the point D fall be B ry/ri • i’ ^ ... XV, ■ ; “4 j / ’ \ Thefctoni Safe* and ABC fall be, a femicircle. , But if the angle ABD be leffe then the angle B AD y then ( by the 2t,*of the firfi) t>pon the right Me B ' Afnidisnio the point in it A } make TpMo the angle ABD an equall angle withiny fegment A B C. Andfo the centre of the circle f allfall my lineD By and it fall be the point E: and the fegment ABC greater then a femicircle Wherefore a fegment the third cafe. beinggeuen yhere is defcribed the whole circle of the fame fegment ; which Was required to'be donfty^ noj of X"; , $&{ Corollary* 1 , » flic '■ nsriT . I Hereby angle is equall to the angle 2) BfJ4ihttt in ajfeQionleJie thenajetnicir - dSrkh afimkfcelejtuffta&r. . .nob %Q i ;Thereis alfoaJTother generall-w'-aytofindcputthe' fore faid centre, which will feme Indifferently for any faction vvhatfoeuer.- And thafis tfius. '^Taicein the cir¬ cumference geuon or {eaion ^AC’fhree pointes at all v , . g sueiitures vvhichller be A,B,C\ And draw thefe lines A B and 'A C(by the firft peticion)' Andf by the 1 o.ofthe firft) deuide into two equall partes either of the fayde iines^he line a^Sinthe point ©p&sthfelii®ei^Cjn tie point E. And (by the 11. ofthefirfl^ifrd'Orth^pointgB^.-'u ' Dd.joj D and id'N.- v f Another more ready Utmonfirfr An ad dition* An other eon- jlructionand demotiflretion of this Propo¬ rtion after , Campari e t&w © and £ rayfe vp vnto the lines and BC perpendicular lyrics *& F and iZF. Map forafmuch as either of thefe angles B D £,and£ EF is a right angle, a right line produ¬ ced frotn the point D to the point £, lhall deuide either of the faid angles : and foraf- much -as it- fallefh vppon the right lines 't> F and % F,it fhall make the inward angles on one and the felfe fame fide, namely, the angles 2) EF and-£ i? F.leife then two right angles. Wherefore fby the fife peticipn) the lines D F and £ F being produced lhallconctirre* Let them concurre in the point F. And forafmuch as a ■ certain e right line D F deuideth a cer- taine right lyneexfF into two equal! partes and per¬ pendicularly, therfore ( by the corollary of the firftof this booke) in the. line D F is the centre of the circle,&: by the fame reafon the cen tre of the felfe fame circle A lhalbe in the right line £ F.Wherfore the centre ofthe . , „ circle wherof ABC is a feftion,is in the point F,which is comrmo fo either of the lines 'D Fand £ F. Wherfore a feftion of a circle being. geue,namely,the fedtion ATS Cthere isdefcnbedthecircleofthe fame fedion: which was required to be done. And-bv this laft generall way, if there be geuen three pointes, fet howfoeuer, fo that they be not all three in one right line,a man may deferibe a circle which fhall paffe by all the faid three pointes. For as in the example before put, if you fuppofe onely the 3. pointes A,ByC, to be geuen and not the circumference A B Cto be drawen,yet follow¬ ing the felfe fame order you did before, that is, draw a right line from *A to £ and an other from B to C and deuide the faid right lines into two equall parts,in the points D and £,and cre<5t the perpendicular lines D £and£ Fcutting the one the other in the pointF,anddrawaiightline from F to £:and making the centre thepointF, and the Space F B deferibe a cirefoan^ it lhall pafie by . the pointes A8cC: which may be pro-? ued by drawing right lines fibm-v? to F,and from F to C. For forafmuch as the two fetes A V zndJ&'F of the triangle ADF are equall to the two fides S TT and £> F of the; triangle cB/D\F(Sax by fuppofitidn the line tAT) is equall to the IineJ9£, and thelyne, £>Fis cpinmontothem both) and the angle A'DFis equall to the angle B D F (for they are both right angles )therfore(by the q..of thefirft Jthe bafe AF ts equall to the' bafe B F. And by the fame reafon the line F C is equal! to the line F B. Wherefore thefe. thred lines F A,F B and F C are equall the one to the other . Wherefore makyng the centre the point. F and the fpace F£,it (hall alfo paffe by the pointes A and C , Which was req aired to be done.This proposition fs very neceffary for many things as you fhal afterward. fee.- -.;r . • .■ ; : ^ '.v^y. . Campane putteth all other way, how to deferibe the whole circle of a feftio geuen . Suppofe that the fc&ion be ^F.It is required to deferibe the whole circle ofthe fame fe&ion.Draw in the fettion twolmes at all aduen- tures AC and B FLwhich deuide into two equal! parts AC in the point £, and B (D m thepointe F.Then from the twopotnteS of the deaifion&draw within the fe&i- ' . ] on two perpendicular lines Ft?. and F H which let cutte- the one the' , other in the.pjmttfc^Mdth-e centre of the circle lhall bein either of the faid perpendicular lines by the cdrollairy of the firft of this booke. Whrirfore the p cle: which was required to be done. But if the lines EG 8c FH do not cucthe one theo- ther, but make one right line as doth 6? Ffinthefecod figure: which happeneth when the two lines AC and B D arc equidiftant. Then the line G H, being applycd to cither part of the circumference geuen, (hall paffe by the centre of the circlc,by the felfe feme Corollary.. H For thelines £ G and F H cannot be equidiftant. For then one and the felf feme circumference ftrould hauc two centres.Whcrfore the line ALC? being deuidedia- hi: to .bC centre ofthccifw l K i; - 1 F C T/V' ' FoLwoe to two equal! partes in the point Kxhx faid point K (hall be the centre of the fe&ion. Pelkaiius here addeth a bride way how to finde out the centre of a circle3which is commonly vied of Artificers, Sup pole that the circumference be A B C D,frhofe centre it is required to finde our. Take a point in the circumference geuen which let be A, vppon which deferibe a circle with what openyngofthecompaileybu will, which let be EFG. Then take an other point in the circumference geuen which let be B, vpon which deferibe an other circle . with the fame opening of the compaffe that the cir- cle EFG was ddcribed,- and let the fame be E H G , which lec'cut the circle E F G in the two pointes E andG-.f I haue not hc_re drawen'the whole circles, blit onely tliofe partes of them which cutthe one theother for auoyding of confufion ) And drawe from thofe centres thefe right lines A E, B E, AG, and B G, which foure lines lhail be equal!, by reafon they are femidiameters of equall circles. And draw a right line from A to B, and fo lhail there be made two Ifofedes triangles A E B,and A G B vnto whom the line A B is a common bale. Now then denide the line A B into two equal partes in the point K which rouft nedes fail betwene the two circumference E P G and EHG, otherwife the partfhould be greater then his whole, Drawe a line from E to K and pro¬ duce it to the point G. Now you fee that there are two Bofceles triangles deuided into foure equall triangles E A K, EBK,GAK and GBE. For the two (ides A Band A K of the triangle AEK are equall to the two fides BE andBK of thetriangleB EK5and the bafe EK is common to them both. Wherefore the two an¬ gles at the point Kof the two triangles A EKand B EKaiebythe 8,ofthe fir ft equall: and therfore are right angles. And by the fame reafon the other angles at the poynte K are right angles, Wberfbre E G is one right lyne by the 14. of the .firft . Which foraf- much as it deuideth the line A B perpendicularly, therefore it pafteth by the center by by the corollary of the firft of this booke. And fo if you take two other poynte s,name« C and D m the cfrcumference gcuen,and vpon the deferibetwo circles cuttyng theone the other in the pointes L and M,ahd by the faid poyntes pro¬ duce a right li ne, it (hall cube the lyne E G beyng produced in the pointe N, w hich ('ball be the cen¬ tre of the circle by the fame Corollary of the firft of this booke,if you imagine the right line C D to be drawen and to be deuided perpendicularly by the lyne L M, which it muft needes be as wc hauc before proned. And here note that to do this me¬ chanically not regardyng demonftration , you neede onely to markethe poyntes where the cir¬ cles cut the one the other,namely, the poyntes E, G,andL,M, and by thefe poyntes to produce the lines ii G and L M till they cut the one the other, and. where they cut the one the other, there is the centre of the circlets you fee herein the feconde figure. "Pd.ij' *A ready way to finde ohs the center of 4 circle co mmo* ly yjed a- mmgft artb- feen. ConjlmUm. Demonftm- nm » 23. 'Theoremc* 1 he 16* Tropofition* Equall angles in e quail circles eonfifl in equall cirmferences * whether the angles be drawen from the centre syor from the circumferences . ^ppofe that thefe circles A 3 C and ID E Ffe equall. And from Mr \centres piamely 3the pointes G and If let there he drawen thefe equal! 'angles 3G C and ELLF: andhkewife from their circumferences thefe equall angles 3 AC and EDF. Then 1 fay3 that the circumference 3 If C is equall to the circumference E LF . Draw (by the firft petition) right lines from 3 to Cj and from E to F. And for afmuch as the circles A3 C and DEE are equally he right lines aljo drawen front their centres to their circumferences „ are (by the firft definition of the third ) equall the one to the other .Wherefore thefe two lines 3 G and G C3are equall to thefe two lines E FI and EL F. And the angle at the poynt G is equall to the an fie at the point LI: Wherfore ( by the 4 *of the firfijt he hafe 3 C isequaMtoybafe EE. And for afmuch as the angle at thepbynt A is equall to the angle at the point D} therefore the fegment 3, AC islilye iq ihe fegment E D F. And they are defcribed W>pon equall right lines 3 C and E F . 3ut like fegmentes of circles defcribed ypon equall right lines } are (by the 24. of the third) equall the one to the other , Wherefore the fegment 3 A C is equall to the fegment EDF. And the ‘whole circle A3C is equall toy 'wide circle DEE . Wherefore ( hy the third common J en tence) the circumference re* mayning 3 JfiC is equall to the circumference: remayning ELF . Wherefore equall angles in equall circles confifi in equall circumferences whether the angles be drawen from the centres or from the circumferences: 'which Seas required to he demonstrated. .... "■ • \ - \ : V : . ' - he %^Theonfae* . ■ -*7- 'Trop.ojttion. In equall circles the angles which conffjn equall .circumfer rences 9ar e equal! the one to the other ? whether the angles he drawen from the centres yor from the circumferences « ■Suppofe * i cfSuclidesElementcs. FoLxou \ yppofij Cfoefe circles AB C,and ID E F, be e quail. Andy>pon thefe equall circumferences of the fame circles yiamely gbpon B C and EF, I let there con ft f thefe angles IB GC and EH F drawen from the cen - tres yindalfo thefe angles BAC and ED F drawen from the cir* cumferences . T hen I fay, that the angle BGC is equall to the angle EHF, and the angle BAC to the angle ED F. If the angle B G C be equall to the an * gle E H Fgthen it is manifelljhat the angle BAC is equall to y angle ED F (by the 20. of the third).But if the angle B G C be not equall toy angle EHF then is the one of them greater then the other . Let the angle BGC be greater And ( by the 21'ofthefirft) ypon the right line BGPand ynto the point ; geuen in it G, c? \ . make Ipnto the , / \ angle EHF \ I j \vY an equall am V J / \\Y gurnet xy \A (by the 26* of b"^~ — — j third') equall \ angles in equall circles confifl fyo equall circumferences whether they be drnwm from the centres or from the circumferences . Wherefore the circumference B ^ is equall to the circumference EF . Bui the circumference EF is equall to the circumference B C: Wherefore the circumference B F( alfo is equall to the cir* eumference BC , the lejfe to the greater : Hfhichis impofiihle , Wherfore the an* gle BGC is not Unequal! to the angle E HE: Wherefore it is equall. And (by the 20. of the third )tbe angle at the point A is the halfe of the angle B G C:and (by the fame ) the angle at the point D is the halfe of the angle EHF. Where * fore the angle at the point A is equall to the angle at the point D. Wherefore in equall circles ,t he angles ^ohich confifl in equall circumferences yare equall the one to the other pxhether the angles be drawen from the centres or from thecircumfe * rences : Ichich leas required to be proued. Demnftr&* tioh leading to an impof- fihlitko $&The 25. 'Thecremc. Ifhe 28. Tropofitlon. In equall circles, equall right lines do cut assay equall cir* cumferences ^ he greater equall to the greater, and the lefie e* quad to the le [te* . S)d.ij. Suppofc Cwflruftm. DemnBra* sim,. The tonncrfe of the former Fropofition. Conflru ction • T>emon$rA~ tmi. F^pofe tfatthefid'rchs ABC*, dndfD M-Ffic efiimll. Andfiiftkefrf hi J there be graven fhefifyuaH tight iikexfB C and EF} Tthichfit cytawpy llthefe dHtmfenntei B A C ahfidbE F beirig ihegrmter }<Malfo fhefo titcimference B& V is eqmfy tithe lejfe circumference E HF. Take (by the fifftof the third) the centres dflhedYctCsfind let the fume he the pointts Efand <L . And dfttw thefe fight fines ? 3fB} Jf€> and L F . And for afmpchps the drcltsrane^diyMrfit^hj thefirft definition of the third) the fines "pBch are dycnvenfro thFcenM' - ■ "■ . . ^ tres are eqtcall .Where* fore thefe two lines (B If mid JfC y are equall to / jg. thefe, two lines* E E and \ p E f. And (by fuppofiti* onjthe bafe EC is equall c, to the ba(W fo •My fir I the angle B'ffC'i's equall to the angle ELF. (But (by the 26* of the third) equal! angles draw en from the centres }conffl tpon equall circumferences, Wher* fore the circumference BGCis equall ti the circumference E H F; andj nhok -Wflt fiJMGAs equall to the fihole circle STEF .‘wherefore the circumference yemqyningB^G, is(by the third common fentenee) equall to the drcumfmnct rmdynirtg ESTF. Wherefore in circles (equall right lin^s do cut away email cfcufifirtncesythegreater equall to the greater ^and thekffe equall to the ieffi: fihichipasrequdredhbeproued. -'"w" ‘ , " "... • ■ .> ' ‘ ■ • • • ■ - i&tfhepretoe. , J ■ f % 1 v ^ ^ :*'t " fi/ ■ M ^UAlickcks finder ^uaUfircumfir^cfis ^efukended enm-mig hefe circles A BC and S) EF fie equall, ytnd in them let there fie tafjm * thefe equall circumferences* 3 B G C and EM Ft anddrawe thefe right lines (B C ana EFfThenlfyythat the right BC is equall to the right line E F . T dke ( by thefirfi of) third ) tlk ’Centres of the circles y anil let them be thepointes Jf and E, and draw thefe rigbtiines If Bj IfC, E EyE F.-, find for dfrhucb as the circuiFferenceB G C is equall ifitlfi circumference EH Fyhe;. angle B:hf 6 is equall tofiangle ELF ( by the 27, of the third) . And for afmucb as the circles jC BC and Sj E F are equalltheone to the Miter yherefire ( by the firfi definitwnofitbe thirdfme lines ' * ’ Tehich ■ jVi * 1 a i FoIaoz* be cen t r es are equall. WhWtfbri- z*'he(ei K-fy are nw, t i-T *V'.» tpejen jjges jf $. and fir jtf they ^ .vi , . f, v , comprehend equal! angles . Wherefore (lip the 4 ♦ dfihefrftjthe'hnfe !BC quail to thebafe EF .Wherefore in equall circles "under equdfl Circulnferekce't^ are fuhtended equal l right lines : yhich yds required to he demon frated. The 4..' Troblenie . The 30. Tfopoftim, ' a circumferencegeuen into Wo eqtialtparUW 1 1 1 is reauired ttfdi* W^yp^Fppofe that the circumference geuen hejiHb3. %ide the:?ircumfererice..A0 %ihib two equall p, artes. (Draw ar is tWfoMiVr'dyfe^fp ZtfitoA'B a perpendicular Unit & And draw thefe right Him A 3D Md0B:Andfbrafmuch as the line A € is equall to the ImeCBfsp the line C 0 is common to them both y there* fore thefe two lines A C and C(D are equal t to thefe two Ims rB C mdC(D.And( by the 4. petition,) the. angle A CD is equal! to the ' cV ’ ds. * - • - ; 7 V X S the, bafeA 0 is effaff. to the b.afe D T$f Bui equall right lines do cut away equall eircum? ferefcespthe greater equall to the greater ftP " a zjgliheoreme. ^Fb.e^u fnaemk m thefommrpk & a.rigfamgfr. *&d]wi* bill Conftmtm . Dmonftr <#- tm r. The firft part nfthtt ThcQ • reme. Smn&pwt* "lUtip&rt. fTbe third fBoo^e but an angle made in the fegment greater then the femieirck islejfethena right angle ,and an angle made in the fegment lejje then the femicircle, is greater then aright angle . Jnd moreouer the angle of the greater figment u greater then a right angle: and the angle of the lejje fegment is lefe then a rightanglc. T K||| P°fi ^dt the cFcle be AB C D,and let the dimetient of the circle be right line BC, and thecetre therof the point R. And take inthefr miclrcle a point at all auentpres ,and lei the fame he (D* And draw theft right lines B A, A C^A Dyxndfp C.Then I Jay that the angle in the femicircle BAC, namely, the angle B AC is a right angle. And the angle A B Clinch is in the fegment AB C being greater then the J'emicircle , is lefe then a right angle. And the angle A ID C -ftbicb is in the fegment A ID C being lefe then the femicircle is greater the a right an « gle.-Draw a line from the point A to the point E,and extend the line B A Imto the point K And forafmuch as the line BE is equall to the line E A, ( for they are dr awen from the centre to the circumference) therfore the an * gleEAB is equall to the angle E B A (by the q.of the firH). Againe forafmuch as the line A E is equall to the line E C, the angle ACE is ( by the fame)equall to the angle C A E.Wherfore the ^ hole angle1. B A C is equal! to thefe two angles ABC and A C B.But the angle FAC lobich is an outward angle of the triangle AB C is (by the 32. o f the firH) equall to the two angles ABCzy AC B. Wherfore the angle B AC is equall to the angle F* A C. Wherfore either of them is a right angle . Wherfore the angle BAC t>bich is in the femicircle BAC is a right angle. And forafmuch as (by the 17* of the firH) the two 'angles of the triangle A 'B C, namely ,A B C and BAC are lefle then tworight angles , and the angle B 'A C is a right angle. 'Therfore the angle AB Cis lefe then aright angle, and it is in the fegment ABC lohich is greater then the Jemkirde . And forafmuch as in the circle there is a figure offourefides , namely, AB C D. But if whin a circle be defcribed a figure offoure fides , the angles therof •tyhich are oppofite the one to the other are equall to two right angles (by the 22» *&f the third) Wherfore (by the fame ) the angles ABC and ADC are equal l . te ofSucliJes Ekmentes . FoLio^ to two right angles SB ut the angle AB C is lefie then a right angle. Wherfore the angle remaining AD C is greater then a right angle , and it is m a figment 'W Inch is kfie then the femicirck \ Now ctlfi I jay that the angle of the greater figment , namely , the angle yghicb is comprehended 'tinder the circumference A B C and the right line A C is greater then aright angle , and the angle of the lefie figment comprehended tinder the circumference AID C , and the right line AC is kfie the aright angle; thick / may thus he proued. Forafmuch as the angle comprehended lander the right lines BA and AC is aright angle, therfore the angle com * prehended -louder the circumference ABC and the right line A C is greater then a right angle : for the ~^bole is euer greater then his part ( by the p. common Jentence. Againe for af much as the angle comprt * bended louder the right lines A C and A F is a right angle , therfore the angle com * prehended lender the right line C A and the circumference ADC is lefie then a right an* gle. Wherfore in a circle an angle made in the femicirck is aright angle, but an angle made in the figment greater then the femicircle is lefie then a right angle, and an an* gle made in the fegment lefie then the femicirck, is greater then a right angle < And moreouer the angle of the greater figment is greater then a right angle: ejr the angle of the lefie figment is lefie then a right angle : lohich mas required to be demonstrated. -A _ M other demonstration to proue that the angle B AC is a right angle. For* dfmuch as the angle AFC is double to the angle B A E(by the ^*cf thefirfi ) for it is equal! to the two inward angles mhich are oppofite. But the inwarde an* gks are ( by the 5. of the fir A) equall the one to the other, and the angle A E B is double to the angle E A C. Wherfore the angles A EB and A EC are double to t he angle B A C. But the angles A E B and A EC are equall to two right an* gks: Wherfore the angle B AC is a right angle. Which mas required to he de* i monSirated. y^Correlarj, fb, • . , ‘ »> - ‘ Her eby 11 is niantfjl fib At if in a triangle one tingle, he etfuall to the two 01 her angles remaining the fame angle is a right The fourth part. The eft USt part* •Another De¬ monstration to prone that theang’e in a femicircle is 4 right angle. A CtmUasy. A*t addition •■affelifatiHS, \ Denton Sira - 1 tion leading So an abjurdi* th i. iAn addition of Cam fane. X . angle', for that the fide angle to that one angle ( namely , the angle which is made of the fide produced without the trian~ gle) is e quail to the fame angles Jaut when the fide angles are equal! the one to the other ft hey are alfo right angles . f yfn addition off’elitarius. v\ V ppofite vnto, the right ' ‘ C ‘ ' ' \\ V : '*.? '• i .' :\. ■ m* • 'c; . : ■ v v 1 \ . ' • A i A.: • ~v Y> v Y If in a circle be infcribed a redangle triangle , the fide o angle fhall be the diameter of the circle. Suppofe that in the circle ABC be infcribed a re&anglc triangle A B C, whofe angle at the point B let be a right angle. Then I fay .that the iide A C is the diameter of the circle . -For if not, then fhall the centre be without the line A C,as in the point E.And draw a line from the poynt A to the point E,& produce it to the circumference to the point D ; and let A E D be the diameter : and draw a line from the point B to the point D.Now(by this' 3 1* Aj Propofitio) the angle A B D flia.ll be a right angle, and therefore fhall be equall to the right angle ABC, namely, the part to the whole : which is ab- furde. Euen fo may we proue, that the centre is in no other where but in the line A C. Wherfore A C is the diameter of the circle : which was required £0 beproued. f Jin addition ofCampane. - - '.v By thys 31. Propofition,and by the 16. Propofition of thys booke, it is mani¬ fold, that although in mixt angles, which are contayned vnder a right line and the .circumference of a circle,there may be geuen an angle Idle & greater then a right angle, yet can there neuer be gene an angle equall to a right angle.For euery fe&i- on of a circle is ey ther a femicircle, or greater then a femicircle, or Idle, but the an¬ gle of a femicircle is by the i<5.of thys booke, Idle then a right angle, and fo alfo is the angle ofa Idle Fedion by thys 3 1 .Propofition : Likewife the angleofa greater jedfion, is greater then a right angle, as it hath in thys Propofition bene proued. he 28. Theorem, The 3 zfPropoftion . If a right line touch a circle ^and from the touch be draymen a nght line cutting the circle: the angles which that line and the touch line maM,are equall to the angles which confjl in the alternate fegmentes efthe circle . Y Vppofe that the right line EF do touch the circle jlECD in the \ point F> ; and from the point F> let there he drawen into the circle \AFCD a right line cutting the circle , and let the famehe FED. " T hen I fay ? that the angles Svhich the line D together Ipith the touch of Smiths Ekmefrte's. Pol . 104,* touch line EF do make , are equallto the angles "Which are in the alternate peg* mentes of the circle ythat is y the angle FBD is equall to the angle "Which conji* fieth in the fegment B A Dy and the angle EBD is equall to the angle scinch conflict!) in the fegment BCD. f aije l)p (by the it, of the frJl)fromy point B » the right line EF a perpendicular line B A . And in the circumference B D take a point at all aduenturcs}and let the fame he C. And draw thefe right lines ADyD C,md CB. And for afmuch as a certaine right line EF tou* cheth the circle A B C in the point B> and from the point B ‘inhere the touch is ra^fedio^nioihe'iouch Im&perpen* ■ dicular B A. fherfore(hy the is?« of the third ) in the line B A is the centre of the circle ABC D.Wherforey angle A D B being in the femictrcle, is(by the Z1* of the third ) a right angle . Wherefore the an* gles remaining BAD and A B Dyare equall to one right angle . But the angle AB Fisa right angle.Wherefore the an* gle ABF is equall to the angles BAD and A BD . F a he away y angle ABD * which is common to them both.Wherefore the angle remayning DBFfs equall to the angle remayning BAD } * which is in the alteimate fegment of the circle . And for afmuch as in the circle is a fgure of power fades ymnelyyA 3 C D)therfore(by the 22* of the third) the angles "Which are oppofite the one to the other yare equall to two right angles. JVberforethe an* gles B A D and BC Dy are equall to two right. angles . But the angles DBF and D B E} are alfo equall to two right angles . Wherefore the angles DBF and DBE^are equall to the angles BAD and BCD. Of "which "We bant pmied that th; angle BAD is equall to the dngle D BF .Wherefore the an* gleremaynmg D B Ey is equall to the angle remayning DCB} "which is in the alternate fegment of the circle- ytamelyjn the fegment D C B.Ifther fore aright line touch a circle yand from the touch be drnwen a right line cutting the circle: the angles rwhich. that line and the touch line make y are equall toy angles "Which cmfft my alternate fegmentes of the circle : * which - Was required to be proued. In thysPropoimon may be two cafes . For the line drawer* from the touch and Siting the circle,, may ey they patfe by the centre or not . If it pafle by the centre, then is it manifeft (by the i8 . 6fthysbooke) that itfalleth perpendicularly vpon the? touch line,and deuideth the circle into two equal! partes, fo that all the angles itf &he feinidrcle,are by the fbrihef Propofitioh, right angles, and therfore equal! to the alternate angles made by thcfayd perpendicular line and the touch linc.Ific paiTe not by the centre, then followe the conftm£tion and dcmon&adon be¬ fore put. '' Cenjbufiiots, Bmortfira- tion. Two cafes in this Prope/i* tisn. Three cafes in this tiopo/i- tion. The firtt (tfe, Conftruftion. Vemon/ka- tioti. Thefecond safe. l&The ^.Trobleme. "The fifProfoJitm. t, v V* . • - '• • r '• rj • i . .« v <? Vppon a right lynegeuen to defcnbe a fegment of a circle » mickfhall contaynean angle e quail to a reBilirie angle geue. $rppofe that the right linegeuen he A Bqand let the reBiline angle gc* iten be C.It is required Tpon the right line geue A Bto defiribe a [eg* ment of a circle "Which Jhall contayne an angle equal! to the angle C -Now the angle C is either an acute angle ? or a right angley ordn ob * tufe angle* Firfty let it be an acute angle as v ■ in the firft defcription.A n d( hy the 23 of the firfi) Wpon the right line A B and to the point in it A defcnbe an angle equal to the angle Cy and let the fame be (DAB* Jf her fore the angle ID Addis an acute angle . From I the point A raife T>p(by then • ofyfrfr) ''mi to the fine AD a perpendiculer line A FL Andfhy the J 9 • of the frjl ) deidde the line A B into two equall partes in the point F. And (by the n. 'of the Jhfiefftofn ihepbintFfaife&p Tmto the line A IB a perpendicular tynq FGfnddYinn dime fromGtoB.Andforafmdchas the line A F is equall to the line F 3 and the line FG is comm on to them both y therfore thefe- two lines AF 'and FG are equall to thefe two lines F B and F G: and the angle AFG is (by the 4g)Ctidon) equall to the angleG F B. JVherfore(hy the 4. of the fame ) the bafe A G is: equall to the bafe G B. Wherfore making the centre G and thefpace GA dejdrtbe {by the 5 .peticion) acircle and it fall pajfe hy the point B : de* fcribefuch acircle & let the fame be A B Be Arid -draw 'a line from E to B.Nop. fora'/imch 'as from the ende of the diameter A E, namely } from the point A is Urawen a right line A D making together "With the right line . A E a right am gkfherfrr/fhythe cor fellary vf the 16. of the third) the line A D toucheth the circle 'A* B"E. AndfotafMmhdsAcertaint "'right line AD. toucheth the circle A1. B Egs from the point A Cohere the touch is (is drawen intoy circle a certaine right line A3: therforefhy the 32. of the third) the angle D A Bis equal! to the angle A EByvhich is in the dltefndte fegment of the cir cle. But the angle DAbB^ is equall to the angle C ^herfore f he angle C is equall to the angle A E B.IVher* fore hpon the right linegeuen A Bis deferibed a fegment of a circle "which com. tayneththe angle AEBfWhkh is equall to the angle geuen /tamely yto C. : Butmufi'ppofe that the angle Cbe a right angle. It is againe required ■pm *** of Sticlides Elements s. FoL port the right line ABto defcribe a feg* merit of a circle , which JhaU contayne an angle equal to the right angle C. Defcribe againe "Upon the right line A B and to the point in it A an angle BAD equal to the reclilme angle geuen C (by the 23* of the fr$l) as it is jet forth in the fecond de* ^ J crip t ion. And (by the 10 ,of the fir ft ) de * ' nide the line AB into two equall partes in the point F. And making the centre the point F and the J 'pace F A or FB defcribe (by l he 3 . petition)} circle ABB. Wher* ** fore the right line A D toilcheth the cir* cle A FB :for that the angle BAD is a right angle W her fore y angle BAD is equall to the angle "Which is in thefegment A E B}for ^Je angk ‘Which is in a Jemicircle is a right angle(by the 31* of the third ) But the angle BAD is equal to the angle C. JVherfore t here is againe defcribe d lapon the Itne AB a Jegment of a circle jUamely }A FBpwhicb contained? an angle equall to the angle geuen namely }to C. But now fuppofe that the angle C be an obtufe angle. Vpon the right tine AB and to the point in it A defcribe (by the of the firft) an angle BAD equall to the angle C: as it is in the third defcription. And from the point Arayjb Wp Tmto the line AD a perpendiculer line A B (by then- of the frjl ) And agayne by the 10 .of the fir ft) deuide the line A B into two equall partes in the point F. And from the point F rayfc lop Imto the line A Ba per pi* dicular line F G (by the 11. of the fame) draws a line from G to B. And now forafi much as the line A F is equal to the line FB} and the line F G is common to them both - th erf ore thefe two lines A F-aiidFG are e* quail to thefe two lines B F and F G : and the angle A FG is (by the 4 ♦ peticion) equall to the angle B FG : "wherfore (by the 4 * of the fame) the bafe AG is equall to the bafe G B.Wherfore making the centre G/ind t he f pace G A defer ibe( by the 3* peticion)a circle and it fhall pafie by the point B: let it be deferibed as the circle AEB is. Andforafmuch as from the ende of the diameter A B is drawen a perpendiculer line AD therefore (by the correllary of the 16. of the third) the line AD touche th the circle AEByr from the point of the touche jnamely, A js extended the line A B. JVherfore (by the 32‘ of the third ) the angle BAD is equall to the angle A HB lohich is in the alternate fegment of the circle. But the angle BAD is equall to the angle C Ee.j. Wher <* E Demnffrs* Thethiri cafe., CmfmUwh Qmenjlrfa mm » Cotijlruftion . Vemonjlra - tmu Wherefore the angle Tohich is in the fegment A HE is equal! to the angle C. Wherfore ypon the right linegeuen A Ejs defcribed a fegment of a circle AH Epvihich contayneth an angle equal! to the angle gotten , namely . C: Tsthich leas required to he done. \ . 4 S^fThe C.Troblems* The 54. . £ Tropofition . From a circle geuen to cut away afeBion which fhal contains an angle e quail to a reBtline angle geuen* j{: Vppofe that the circle geuen he AC and let the reHiline angle geuen ^y^r.be D. It is required fro the circle A 3 C to cut away a fegment "Which fall contayne an angle- equal! to the angle D. Draw(bythe 17 of the / third) a line touching the circle, and let the fame be EE: and let it touche in the point 3. And (by the 23. of the prfi) hpon the right line EF and to the point in it 3 defer ibe the angle F EC equal l to the angle ID. ISiom for aj much as a ccrtayne right line E F touche th the cir* cle A3 C in the point 3: and foray point of the touche ynamely fB } is drawn into the circle a ckrtaine right line 3 C , therefore (by the 32* of the third)the angle FBC is equal! to the angle 3 yfC Tvhich is in the alternate fegment. But the angle F 3 C is E equal! to the angle D. Wherfore the angle 3 AC Tehich con file th in the fegment 3 A C is equal 1 to the angle 55. Where* fore from the circle geuen A 3 C is cut away a fegment Eg! C gtobich containetb an angle equal 1 to the reHiline angle geuen: Hitch Teas required to be done . \ ^ a* — j> ) v \ ' 1 ' • -) ■ ' T* he zp . Theorems , The ffropojition • If in acircle two righ times do cut the one the other Jtherect* angle paraUelograme comprehended vnder the fegmentes or parts of the one line is equaU to the reBangle paraUelograme l comprehended vnder the fegment or partes of the other line . '■ ' V- . ; : . . ,, ' . ,, « I: . ", - - ; .» '» , y ; \\ ' •. v '* - "4- Etthe circle be A E C Dy and in it letthefe two right lines AC and E D cut the one the other in thepoint E.Thenlfay that the reHangh farallelogramme contayned lender the partes A E and EC is equal! to the ofSuclitles Elementes , Fq/aq6, be drawen by the ce ntre /hen is it manifeji fhat for as much as the lines A E arid E C are e quail to the lines © E and E B by the definition of a circle } the re cl angle parable lograme alfo contayned Imder the lines A E and E C is equall toy reft angle paralle * lograme contained bnder the lines IDE and ED. But now fippofe that the lines A C and ID B be not extended by the centre ymd take(by the i. of the third) the centre of the cir* cle AD C D}and let the fame be the point F, and from the point F draw to the light lines AC and © B perpendicular lines FGand F H. (by the n, of the fir ft) and draw thefie right lines F BfF C pnd FE. Andforafmuch as a certaine right line FG drawen by the centre yutteth a certaine right line AC not drawen by the centre in fuel? forte that it rnaketh right angles } it therfore deuideth the line A into two equall partes (by the 3. of the third). Wherfore the line AG is equall to the line GC. And foraf much as the right line AC is deuided into two e* quail partes in the point G} and into two Unequal! partes in the point E: therfore ( by the 5 • of the fecond ) the re ft angle paralleled gramme contained lender the lines AE and E C together "frith the fquare of tf?e line E G is equall to the fquare of the line G C. Tut the fquare of the line G F common to them both pfrher fore that "frhich is contained Imder the lines AEtsr E C together -tyitb thefquares of the lines EG and G F is equall to thefquares of the lines GFEr G C. But l?nto y fquares ofy lines EG iy GF is equally fquare ofy line F E (by the 4-1. of the frit): and to the Jquares of the lines G C andGF is equall the 'fquare of the line l C (by the fame) Wherfore that 1 frhich is contain nfdynderthe lines A E and E Cpogether with the fquare of the line F E is e* quail to the fquare of the line FC. But the line F C is equall to the line FB. For they are drawen from the centre to the circumference. Wherfore that "frhich is contained hnder the lines A E and E C together 1 mb the fquare of the lyne FE is equal to the fquare of the line F B.And by the fame demonftration that "frhich is contained lender the lines © E and E B together "frith the fquare of the line F E is equall to the fquare of the line F B. Wherfore that "frhich is contained "bn* der the lines A E and E C together -frith the fquare of the line E F is equal l to that - frhich is contayned Imder the lines © E and E B together " frith the fquare of the line EF.T ake away the fquare of the line EE "frhich is common to them both. Wherfore the re ft angle par allelogramme remayning -frhich is contayned Imder the lines A E and E C is equall to the reftangle par allelogramme remay * idngj which is contayned bn der the lines © E and E B. If therefore in a circle two fight lines do cut the one the other : the reftangle par allelogramme compre* Ee. tj , bended the reftangle parallelogramme contained frnder the partes ID E and E B.For if the line A C and B ID Two Cafes its this fiopo/i- tion , Ftrft cafe* Detnonftra* tion. The fecond C*f<?r tonfruetkn * Vemotiflra - s'wu Three cafssin this I’ropQj'i- than . The third safe. -i 7 he ttiirdTfooke hendedlander the fegmentes or parts of the one line is equall to the reBan gle pa$ rallelograme comprehended lender the fegmentes or parts of the other liner^hich "'was required to he demonUrated. In thys Proportion are three cafes : For eyther both the lines paffebytheceiv tte, or n eyther of them paiTcth by the centre : or the one paffeth by the centre and tire other not. The two firlb cafes are before.dcmonilrated. •*- 7" But now let one of the lines onely, namely, the line zAC pafle by the centre, which let be the poynt F, and let it cut the other line, namely, B E, in thepoynt E . Now then the lind AC deuideth theline BE eyther into two equall partes,, or into two vn- equall partes . Pyrft let it deuide it into two equall partes :Whcreforefa!fo it deuideth it right angled wyfe by the 5. of thys booke . Drawc aright line from B to F. Where¬ fore B EF is a right angled triangle . And for afmnch as the right line AC is deuided into two equal! partes in the poynt F,& into two vnequall partesjin the poynt £ . Ther- fore the .redangle figure contayned vnder the lines zA E and E C together with the fquare of the line E f,i $ equall to the fquare of the line F C (by the 5. of the fecond). But v.nto the fquare of the line FC is equal! the fquare of the line 2? F /for that the lines F B and F C are equall). Ther- fore that which is cctayned vnder the lines AE and E C together with the Square of the line E F, is equall to the fquare of the line B F, Butvnto •the Square: of th'ciinc B'F, are equall the fquares oftheiints BEzndEF (by the 47. of the hrft). Wherefore that which is contayned vnder the tines %AE and EC together with the fquare of theline E F., is equall to the fquares of the lines ME and EF. lake away the fquare of the line , - EF which is common to them both ; Wherefore that which remayneth, namely, that which is contayned vnder the lines *A £ and EC, is equall to the refidue, namely, to the fquare o f t he line BE. B ut the fq uare of the line B £ is that which is contained vn¬ der the lines B £ and E E for (by fuppofition) the line BE is equall to theline E E, Wherefore that which is contayned vnder the lines A E &£C,is equall to that which is contayned vnder the lines BE and E E s which va$ required to beproued. But uowlet the line zA Cpaffing by the centre, deuide; the line B D notpaffing by the centre, vn- equalW in the poynt E . And fro the poynt £ raife vp vnto the line &AC a perpendicular line E H, which produce on the other fide to the poypt G. Wperefpre (by the 3. ofthisbooke) the line EH is equall to the line E G . Whertore as we haue be¬ fore proued , that which is contayned vnder the & lines A E and £ C, is equall to that which is eon- tayned vnder the lines GE & E H ; but that which is contayned vnder the lines B E and £<Zhisalfo < equall to tbft which is contayned vnder the lines G E and £ H, by the fecond cafe of thys Propositi¬ on : Wherfore that which is contayned vnder the lines e^£ and £Cris equall to that which is con- ■ ,v tayned vnder the lines BE and EE 1 which was agayne required t? be proued. Amongeftali the Propofitions in this third booke3doubtles thys is one of the chiefeft . For it fetteth forth vnto vs the wonderful! nature of a circle . So that by « ■ 7 *• r. w - it '7 - M . _ ■ \ \ _ \ F • - E / \ ■ c m v ofSuclides Element es . Foh\ojt it may be done many goodly conclufions in Geometry , as (hall afterward be de¬ clared when occafion lhall feme. ’W '* •* " * '* ' * ' ' £yThef o. Theorem e. The %6. Tropojition* * ■ - ' * ■. \\ , if. \ - .. , 'V:. •' * ■ . .. .. •- -■ \ > -•_ / _• If without a circle he tahgn a certaine points and from that point he drawen to the circle two rift lines fo that the one of them do cut the circle ? and the other do touch the circle: the rectangle parallel ogramm e which is comprehended "snider the whole right line which cutteth the circle, and that portion of the fame line that hetb hetwene the point and the vttercir > deference of the circle , i s equall to the fquare made of the line that touched the circle. * •' - . . > ei ■% i 2d. . Vppoje that the circle be AB C : and without the fame circle taken* ny point at all aduentures/nd let the fame he CD, And from the point ! D let there be drawen to the circle two right lines DC A and D and let the right line DC A cut the circle AC Bin the point C /aid let the right line B D touch the fame. 'Then I jay } that the reftangle parallels • gramme contayned "tinder the lines AD and D C, is e quail to the fquare of the dine BD . IsLow the line DC A is either drawen by the centre /r not . Fir ft let it be drawen by the centre . And (by the frft of the thirdjlet the poynt F hey centre of the circle ABC } and drdwe a line from F to B< Wherefore the angle FBD is aright angle. And for afmuch as y right line A C is deuided into two equall partes in the poynt F/nd "tinto it is added directly a right line C D /her for e( by the 6 . of the fgcond ) that 'Svhicb is contayned "Snider the lines A D and D C together yoith the Jquare of y line C F, is e quail to the 'Jquare of the line FD. But the Ime F C is e quail to the line F B} for they are drawen from the centre toy circumference: Wher * fore that lohich is contayned ". tinder the lines AD and D C together t>ith the fquare of the line FB, ts equall to the fquare of the line FD.Buty fquare of the line FD> is (by the 47. of thefirftj equall to the '/quarts of the lines F B and BD (for the angle FBD is a right an fee) . Wherefore that which is contayned "tinder tfa lines AD and DC together With the Jquare of the line FByis equall to the Ee.iij. j quarts CotlftyuakXi Two cafes in this Propofi - tiov. ‘IbefirRcapte DmmUra- Tktf?c9nA stft, Conflmtim ♦ VemonffM* eion. fquares of the lines F $ and T 0 . Take away the fquartof the lineFTidhkh k common to them both . Wherefore thatlohich remaynethy namely, that lohicb is contayned lender the lines AT) and D Cy is equall to the fquare made of the line D B lohichtoucheth the circle. Tut now fuppofe that the right line D C^Ahe not drawen by the centre of the circle ATC. And (by the fir ft of the third) let thepoint E bey cen <* tre of the circle fifT C. And from) poynt Efiraw (by the 12. of the fir ft) Tmto the line AC a per* pendicular line EE y and draw thefe right lines ETjECynd ET) .TSLow the angle EE ID is a right angle .Andforafmuch as a certaine right line E E drawen by the centre yutteth a certayne E other right line AC not drawen by the centre fin fuch fort that it maketh right angles 3 it deuideth it(byy third of the third) into two equall partes. Wherefore the line A F is equall to the line FC. And for afmuch as the right line A C is deuided into two equall partes in the poynt Fyi? 7>nto it is added directly an other right line making both me right line y therefore (by the 6 . of the fecond) thatlohich is contayned Crider the lines Djland D C together ‘With the fquare of the line E Cy is equall to the fquare of the line F 0 ; put the fquare of the line E E common to them both. Wherefore that nhich is contayned Imder the lines 0 A and 0 C together 1 vith the Jquares of the lines C F andFEy is equall to the fquares of the lines F 0 and EE. Tut to the fquares of the lines F0 and F Ey is equall the fquare of the line DE( by the 4.7 -of 1 the firfl )for the angle EE D is a right angle . „ And to the fquares of the lines C F and EE, is equall the fquare of the line CE (by the fame). Wherfore that ahich is contayned Tut* der the lines A 0 and 0 C together 1 nth the fquare of the line EC fits equall to the fquare of the line ED . Tut the line EC is equall to the line ET: for they are drawen from the centre to the circumference . Wherefore that lohicb is con « tayned lendei the lines A D and 0 C together loith the fquare of the line E Ts is equall to the fquare of the line ED . Tut to the fquare of the line EDyaree* quail the fquares of the lines E T and TD( by the 47. of the firSl)for the an* gle ETD is a right angle : Wherefore that lohicb is contayned lender the lines A D and DC together loith the fquare of the line ETfis equall to the fquares of the lines E T and BD .Takeaway the fquare of the line ET lohich is com* mon to them both: Wherefore the refiduey namely y thatlohich is contayned 'ion* der the lines jiD and D Cy is equall to the fquare of the line D T . If therfore loithout a circle be taken a certaine point y and from that poynt be drawen to the circle two right lines yfo that the one of them do cut the circle f and the other do ofSttcliJes Elemenies. touch Whclrcle i We. reftangk pdralhlogramme isMch is Comprehended Tmdef the lehole right fine uhichcutteththe circle and that portion of the fame line that Ueth betwene the poynt and the latter circumference of the circle js equall to the fquare made of the line that touche th the circle : lohich leas required to h demonftrated . * % A | f ... ■_ \] n t ; *!. , 1 ' ’•."r V* - • - - fT ipo Corollaries out of Campane ■* If from one andthe felfe fime poynt iakfn Without a circle be drawen into the circle lines hov» many footer : the retiangle Parallelogf amines contaynedvnder every one of them and hys outward are t quail the one to the other. if . " ' 4 _ . •- ■ _ - ' • •" - , ■ ' And thys is hereby manifcfl, for that euciy one of thoie redangle Parallelo- grammes arcequali to the fqtrare of the line which is drawen from that poynt and toucheth the circle by thys 3d* Proportion . Hereunto he addeth* If two lines drawen from one and the feif: fame point do touch a circle, they are equal! the one to the other. * Which although it neede no demcnftration, for that the fquarc ofeyther of them is equall to that which is contayned vnder the line drawen from the fame poynt ana hys outward part : yet he thus proueth it. Suppofe that there be a circle B CD, whole A centre let be E , and without it take the point A „ And from the poynt A drawe two lines AB and *AD, which let touch the circle in the poyntes B and D . Then I fay , that they are equal! . Draw* thefe right lines EB,ED, and AE . And by the l8. of thys booke,eyther of the angles at the poyntes B and® is a right angle. Wherefore (by the 47. of the frit) the fquarc of the line E, is equall to the two fquares of the lines A B and EB : and by the fame reafon,to the two fquares ofthelines AD and ED . Wherefore the two fquares of the lines AB and EB,axt equall to the two fquares of the lines tA D and E D.And rforafmuch as the fquares of the lines EB and ED are eqiiallrthereforethetwo other fquares ofthelines AB and AD are alfo equall .Wher- fore the line AB is equall to the line A D ; which was required to be proued. The fame may be proued an other way : Draw a line from B to D, And (by the $ .61 the firft) the angle EB D i s equall to the angle E D B . Andforafmuchasthe two an¬ gles A B E and <tA D E are equall,naniely,for that they are right angles : if you take from them the equall angles EB D & ED B, the two other angles remayning, namely, the angles ABD and ADB flialTbe equall . Wherefore( by the <5. of the firft} the line AB is equall to the line A D. \ /\ \ I • E ) 1 / ! / . y f Hereunto alfo Telitarius addeth this Corollary. Trim a poynt geuen Without a circle, can be drawen vnto a circle onelytivo touch lines . The former defeription remayning, Ifay that from the poynt A can be drawen vnto Ee.iiif the Fits! CortHa- Second Co* tddttry. Third Cord* lary, • thecircle BCD no more touch lines,but the tsvo lines’ A B and AD. Forifiebepof- iible, let A F alfo be in the former iigurc a touch line,touching.the circle in the poynt F.Andprawe a line from E to F. And the angle at the point F fhall be aright angle, by the 1 8 . of this booke : Wherefore it is equail to the angle E B A> which is contrary to the 20. of the firir. This may alfo be thus proued. For afmuch as all the lines drawen from one and-the felfe fame poynt & touching a circle are equail, as we haue before proued,but the lines 3 and AF can not be equail, by the 8. Propofition of this booke, therefore the line *A F can not touch tfe circle BCD „ y&ffhe iiSIheoreme* 1 he 37. Tropofition* If without a circle be taken a cert nine point , and from that point be drawen to the circle two right lines jf which y the one; ■ doth cut the circle and the other falleth vpon the circle , and i ha t in fuch fort }t hat the rectangle parallelogramme which is cotayned 'under the whole right line which cutteth the circle s and that portion of the fame line that heth betwene the point and the y tiercircumferece of the circle fits equail to the (quart made of the line that falleth ypon the circle : then that line that jo falleth vpon the circle fhall touch the circle. "This propor¬ tion ts the co¬ ney/ e ofibe former. Confrufmn, Vemonftra- r Et the circle be ABC: andlcith* Of out the fame circle take a point } and » the fame be ID, & from the point fffiffD let there be drawen to the circle ABC two right lines ID C A and D B 2 and let DC A cut the circle } and D B fall ypon the circle, And that in juch Jort} that that fhich is contained lender the lines AD and DCy be equail to the fquare of the line DB . Then I fayjhaty line D B toucheth the circle ABC, Drawe (by the 17* of the third ) from the poynt D a right line touching the circle A B C } and let the fame be DE. And( by the fir/1 of the Janie ) let the point F be the centre of the circle ABC: and draw thefe right lines FE,FB, and F D . Wherfore the angle FE D is a right angle . And for afmu